In a small bowl, stir together the mayonnaise and minced garlic; refrigerate until ready to assemble.
Preheat the oven to 375°F (190°C). Unroll the refrigerated pizza crust onto a 15×10-inch jelly roll pan and press to fit the pan.
Bake the crust for 10 minutes, then remove and let cool slightly; cut the baked crust into 4 equal rectangles.
Heat 1 tablespoon of the olive oil in a large nonstick skillet over medium heat. Add the sliced red onion and sauté 3 minutes.
Add the granulated sugar, balsamic vinegar, and crushed red pepper flakes to the onions and sauté another 5 minutes until soft; transfer onions to a bowl and set aside.
Wipe the skillet clean. Flatten each chicken breast to about 1/4-inch thickness using a meat mallet or rolling pin; season both sides with salt, pepper, and dried thyme.
Heat the remaining 1/2 tablespoon olive oil in the skillet over medium-high heat. Cook the chicken 4–5 minutes per side, until cooked through; transfer to a cutting board, rest 5 minutes, then slice.
Assemble 2 sandwiches by spreading the mayonnaise mixture on one rectangle each. On the other 2 rectangles, arrange sliced chicken, squeezed spinach, and sautéed onions, then top with provolone. Place the mayonnaise-spread rectangles on top, mayo side down.
Lightly grease a cast iron skillet with a bit of olive oil and heat over medium-high. Cook sandwiches, pressing gently with a spatula, until golden brown on the bottom, then flip and brown the other side.
Cut each sandwich in half and serve warm.