How to expand a pulse by a factor
Rather than doing this with brute force and a GHz-clocked FPGA, this sort of job is well suited for a so-called Time to Digital Converter.

The basic concept is to integrate the pulse in the analog domain, charging a capacitor to a voltage proportional to the duration of the pulse.  Then sample the capacitor voltage with an ADC, discharge the capacitor (actively or passively) and repeat.

There are ICs that will do most or all of this for you, or you can build one from a a couple of fast op amps, an ADC and some passives.

will try to flesh this out further with examples/links when back at a real keyboard