Hi,
Are we talking the same kind of ASIC? The ones we used back in the 1980's were programmable, such that they programmed similar to how PROM's program, by blowing out tiny "fuses" in the die. Of course you might be talking about making the chips themselves before they are programmed.
The chips came with a bunch of logic and flip flops, and they were connected in various ways. But to get them to function the way we wanted them to, we'd have to figure out what fuses had to be blown out. Once done, the flip flops could be arranged into a string of binary counters or BCD counters or whatever, along with logical gates connected in various ways due to the fuse blow outs.
There were different part numbers which had different overall functionality so one may have more logic gates or flip flops then another. You had to get a part number that had enough flip flops and gates to do the job you needed. But the parts were always the same not custom. They were only custom after being programmed, just like a PROM. So it was never a matter of making an entire custom IC chip.
It's been so long now though i hardly remember any details.