This surely ends up in a complex circuit comprised of several ECL MSI devices. Onsemi has a large portfolio of ECL ICs, also including complex devices like counters and shift registers.
With pulses this narrow, I would design it from the output socket, backwards and just use the micro as a controller of switches and nothing more. If you build a delay line with varactor capacitors, then you can electronically control the capacitance and hence the delay, gating the input pulse with the output pulse should get you a variable and very narrow output pulse (use ECL).
Frank
Using analog circuits for the pulse generator is still an option. But you can achieve about 100 ps timing resolution of adjustable pulsewidths using recent low cost FPGAs, together with effectively unlimited waveform complexity. There will be always room for analog solutions of special problems. But once you have learned to manage the digital methods, you'll appreciate their performance.