The only carbonara I remember making was at a restaurant:
handful chopped prosciutto
cup or so heavy cream
an egg yolk
grated Romano
linguine
chopped flat leaf
Saute prosciutto - mix egg yolk with cream and add - bring to simmer before adding cooked linguine - toss in Romano, lots - plate and sprinkle green stuff on the white stuff.
Sauce was heated, not reduced. The cheese helped thicken it, but it was served a little bit soupy.
I didn't know that the cream was NOT traditional. See, I learned something today.