Toast pepper, coriander and cumin together in a dry pan until they crackle and pop (shake pan constantly). Their color should be light brown, and spices should smell like toast, popcorn, smoky, etc. Pour them straight into a coffee grinder and blend the seeds to a fine powder. Mix all spices, garlic and salt well into the lamb. Form into 6 even patties.
After weighing, rip the dried peppers open to shake out the seeds (a few seeds left in is ok). Throw the peppers in a saucepan with the water, and boil over medium heat until only 1/4 cup of liquid remains at the bottom. Add the peppers (with cooking liquid), sour cream, oil, garlic and lime to the blender, and blend it until very smooth. Add salt and blend again. Add cream and pulse a few times just to incorporate and get a saucy, squirtable consistency.
Grill the lamb patties and the ciabatta until nice, dark grill marks are everywhere (lamb is at its juiciest around medium rare). Put about 1 teaspoon of sauce on each side of the bun. Slice the patties in half to lay lengthwise on the bread, and crumble Cotija cheese on top. Dress with plenty of juicy tomato slices, onion and lettuce. Cut sandwich lengthwise (along the seam of the lamb patty) and plate with cross-section facing outward.