Preheat oven to 425 degrees. Lay prosciutto in a single layer onto a large rimmed baking sheet. Bake in preheated oven for 5 minutes, flip slices over, continue cooking another 5 minutes until crispy; remove from oven, set aside.
Peel and remove seeds from cucumber; shred cucumber and squeeze dry. Place into a small bowl, add in the yogurt, feta cheese, 1/4 teaspoon salt and 1/8 teaspoon black pepper; mix to combine; set aside.
In a medium bowl, place the tomatoes, capers, lemon zest, lemon juice, olives, basil, 1 1/2 tablespoons Ouzo, and 1/4 teaspoon black pepper; toss to combine; set aside.
In a large bowl, place the ground round, garlic, onion, Greek peppers, dried oregano, remaining 3 tablespoons Ouzo, remaining 1/4 teaspoon salt and remaining 1/4 teaspoon black pepper; mix just until combined, do not overwork the meat. Form into 4 patties, slightly larger than the pita bread; set side.
Heat a large non-stick grill pan or skillet over medium heat. Add in the burgers and cook for 3 minutes; flip burgers over, squeeze the 1/2 of lemon over burgers, top burgers evenly with fontina cheese and continue to cook until burgers are cooked through and cheese has melted, approximately another 3 minutes.
To assemble, spread approximately 2 - 3 tablespoons of the tzatziki sauce onto the bottom half of each pita bread, top with a burger, 1 1/2 slices of the prosciutto, tomato/basil salad, then with the top of the pita bread. Serve with remaining tzatziki sauce on the side.