<I apologize for being so verbose.>
One of the best reads I have had this year online on any subject, thanks to everyone. I love the range of feelings and advice also, I think it is reflective of different age ranges, but also of different life goals and experiences. What is odd is that I agree with so many opinions, even those that seem contradictory. I definitely felt like this poster did earlier in my career, right or wrong...
At least in the USA, and probably true in most first world countries, some people get obsessed with the Joneses and chasing the dollar. I am currently in my highest paid position by a large percentage, and am compensated well above market value. I was very fortunate to find this opportunity soon after I had to shutter my own consulting business due to the economic crash. However, I really do not like the work- but I do like my coworkers, my schedule, flexibility of my bosses, and some other aspects of the job. It has allowed me to catch up on my 401k a LOT, and put my family in a better overall financial position. However, even if this position could last forever, I cannot keep doing this forever. I don't really like living where do (relocated for the job), and the work has become a grind. I have enjoyed other jobs much more for less pay than my current one, but I wasn't in as good of an overall financial situation. There are tradeoffs to most things in life.
I think engineers for some reason come into this career-game hardwired to WANT to really enjoy what we do, and compensation to us may seem secondary. And when we do not enjoy what we do, it really bothers us. I happen to have several friends who are very well-compensated attorneys, and almost none of them like what they do, and a couple of them have admitted hating it. The extra Porsche in the garage only buys so much extra happiness I guess (literally 2 of them have an extra Porsche in their garage).
Specifically to you being a young engineer, I think it is wise to stay put for a minimum of 3+ years out of college as others have said. Think of your career like a 40 year contract, and it is back-end loaded. You are in the minor leagues now, and you are getting better every day, and getting experience, and you will get a better contract down the road. Being a good engineer is not just about intelligence and ability, or putting in overtime, but a huge chunk of it is EXPERIENCE, and it is invaluable. Keep building your resume, and honing your skills.
HOWEVER, this is a capitalist society, and like it or not, we are all commodities. Do not compare your salary to other industries or occupations, that is a fool's errand and will only frustrate you. If I could have been born with an arm that can throw a ball 100mph I would not be on this damn message board right now. Also, you WANT your bosses to be rich!! That means they know what they are doing and you are working for smart, successful people! What is the alternative? A dumb and poor boss? Poorly run company? How would that work out for your making more money? This may be the one of the most irritating remarks many people make when they grumble about their current state of employment (I am not aiming at you), though I get it is a pretty normal emotional reaction, you have to be more analytical (engineer brain).
BUT, you could compare yourself to peers in your field with your experience level in your locality, which is likely what your company is doing if they want to retain employees and also not overpay them (market analysis). Can you evaluate your own competence fairly and compare that with peers in your discipline? It is not just about years on the job and hours per week worked, though that does count for something. How much skill, knowledge, and experience do you have compared to similar engineers in NYC? Do you get projects done on time, under budget, with minimal mistakes? How are your people and client skills? All these things may be part of what your boss is considering.
You may find most peers are within +/- 5% of your total compensation (include time off, medical/dental, bonus, etc.). Now- put value on your mentors, coworkers, and the overall type of work you do and your experience. This is very difficult, but try to put a value on all of that. Remember this is an investment in your future, like a bond that will mature in 10+ years. If you have a couple of go-to mentors and you are really learning fast under their wing, put a lot of value on that! You may find even if you are being paid a little under market, the experience you are getting is above market value. This will get you paid more down the road than lesser-valuable experience would.
So, looks like I am about to turn 40 and now I think a lot like the rest of the "old guys/gals" on this board. Oh well, getting old ain't so bad... I would stay put for at least another year or so, even until you get your PE, especially if you are getting good experience on quality projects, and they are giving you progressive responsibility. Don't compare yourself with the Joneses, especially in other occupations. However, do the analysis in my above paragraph if you think you are being grossly underpaid. You can always interview somewhere, get an actual offer, and then go to your boss and CAREFULLY have a discussion about YOUR market value. "I really enjoy working here and love the experience I am getting, but I must also consider my market value and what is best for my family. Is there anything that we can do?" If your boss balks, you have a backup plan.
PS
At a minimum put away what your employer matches in your 401k (usually 3% or more). If not, you are passing up free money. Also, every dollar you put in your 401k is a dollar that does not get taxed (yes, you will pay tax eventually), so it is like making 15%+ interest off the top. Save until it hurts. Nobody gets to 50 or 60 and says, "Wow, I saved way too much, that was so stupid!" You are an engineer, so you understand numbers- compound interest is a powerful tool. Use it to your advantage early in your career.