You are right, there's very little fake cheese in our house except for the smoked plastic cheese
(Tesco Austrian Smoked Processed Cheese 125G)

My husband
loves that stuff! But then he found a wheel-shaped one that he likes even better...
But I digress. You are describing this :

PRÉSIDENT
EMMENTAL SWISS
Pressed, hard cheese with distinctive holes and nutty, sweet flavor. Made from pasteurized milk of grass-fed cows in the northwestern region of France
A huge favourite of mine.
It melts well? I will get some and try this. Now I need to get some thick-sliced soft sourdough (I don't even like sourdough!) to put this all together. Oh. And I think there was a smear of garlic mayonnaise on it. I remember some kind of tasty goo...
One day, I'm going to tackle Panini's overstuffed sandwich... but that's a quintessentially American thing, with all that processed meat on it, I think. But one day... *wistful sigh*