Heat the oven to 250°F. Heat the oil and garlic in a 10-inch heavy-duty skillet (preferably cast-iron) over medium-high heat until the garlic starts to sizzle steadily and browns in places, about 2 minutes. Add the spinach, sprinkle with 1/4 tsp. each salt and pepper, and cook, tossing, until just wilted, about 2 minutes. Transfer the spinach to a colander. Let cool a couple of minutes, discard the garlic, and gently squeeze out the excess liquid from the spinach.
In a medium bowl, toss the spinach with the fontina, parmigiano, sun-dried tomatoes, and 1/4 tsp. pepper. Set 3 of the pita halves on a work space and top each with 2 slices of the prosciutto. Top each evenly with the spinach mixture and set the remaining 3 pita halves on top.
Wipe the skillet clean with a paper towel and heat the pan over medium heat. When the pan is hot, add one of the pita sandwiches and set another medium heavy skillet on top of the sandwich. Put 2 lb. of weights (cans work well) in the empty skillet and cook the sandwich until the bottom starts to brown, about 2 minutes. Flip and cook the other side until it browns and the cheese starts to melt out the sides, about 2 minutes. Transfer to a baking sheet and keep warm in the oven. Cook the remaining sandwiches in the same manner. Cut the sandwiches in wedges and serve.