"The objective is: given a first prime, generate the $n$ next primes."
This is equivalent to, "given an integer $N$, find the smallest prime exceeding $N$."
People do this all the time, for example, here is a tabulation of the smallest prime exceeding $10^m$ for various values of $m$. But there's no especially clever way to do it. In effect, you look at $N+1,N+2,N+3,\dots$ until you find a prime. You can save some work by sieving, but you already know that. If you want to find, say, the first 50 primes after $10^{100}$, no doubt you'd save a lot of work by doing some sieving, but in the end there would be a lot of numbers that get through the sieve, and you'd have to fall back on the standard primality tests to see which ones are actually prime.
I guess the other thing you can do is use the Prime Number Theorem to estimate how far you have to sieve to have a good chance of catching the next 50 primes. Roughly speaking, one number in every $\log N$ will be prime, so if you go out to $N+100\log N$ you should have an excellent chance of catching the next 50 primes.