Preheat the oven to 415°F and place a rack in the middle position.
In a small bowl, combine 1/2 cup creme fraiche with 1 tbsp chunky garlic (or 2 cloves pressed) and 1 tsp fresh oregano (or 1 tsp dried). Stir until evenly mixed.
Place the 3 pieces of pita bread (or flatbreads) on a baking sheet or pizza pan, spaced so air can circulate between them.
Evenly spread the creme fraiche mixture over the top of each pita, leaving a small edge if you prefer a crust.
Divide the 1/4 cup freshly grated Parmesan and 1/4 cup freshly grated Asiago evenly over the three pitas.
Distribute the 4 cups of fresh baby spinach leaves evenly over the pitas (pat spinach dry first if it is wet).
Top each pita with the 8 oz sliced fresh mozzarella, arranging the slices so the cheese covers the spinach.
Bake for 12–15 minutes, until the mozzarella is bubbly and lightly browned. Remove from the oven, let cool slightly, then cut and serve.