I first had sabich almost a decade ago at an Israeli restaurant near me. I got it as a plate with chopped, roasted (I think) eggplant, hummus and Israeli salad (diced tomato, cucumber, green bell pepper), a hard boiled egg and a thick, warm, fluffy pita. I made pita sandwiches out of it and I loved it; their pitas are excellent. The hot sauce accompanying it I judged to be harissa since I've never heard of schug but I later saw it elsewhere on the menu and found the Sabra product in the Kosher section of a nearby grocery store. The Sabra schug (red or green) is a thick, rather dry paste-like product but what I had at the restaurant was flowable. I've also had it there as a sandwich on a baguette, with thick slices of eggplant plus tomato and cucumber.and hard-boiled egg They also offer it as a pita sandwich and a wrap with malawach, the fried Yemeni bread similar to porotta but made with phyllo, I think. I see places online giving the pronunciation as sabikh but that place said it was sah BEECH. So far as I can recall, there was no amba. I'll have to look for that at the grocery store.
I see another Israeli restaurant nearby has added it to the menu also as a pita sandwich, on a baguette and as a wrap in lafa, the Iraqi flatbread. Their lafas are so big I probably wouldn't be able to finish one at one sitting.
Serious Eats did a recent feature on sabich and I agree, it's far superior to falafel.