In a small bowl, stir together mayonnaise and garlic. Refrigerate until needed.
Preheat oven to 375 degrees.
While oven heats, sauté the onion in the olive oil in a large nonstick skillet for 3 minutes over medium heat. Add sugar, balsamic vinegar, and crushed red pepper flakes. Sauté another 5 minutes. Set aside.
When oven reaches 375 degrees, unroll the pizza dough and press into a 15×10-inch jellyroll pan. Bake for 10 minutes. Let cool and then cut into 4 rectangles.
After onions are cooked, transfer them to a bowl. Wipe the skillet clean. Flatten the chicken breasts with a meat mallet or rolling pin. You want them to be about 1/4-inch in thickness. Season with salt, pepper, and thyme.
Heat 1/2 tablespoon of olive oil in the wiped-clean skillet over medium-high heat. Add chicken and cook for 4 to 5 minutes per side or until cooked through. Remove to a cutting board and let cool 5 minutes. Then cut into slices.
To assemble, spread the mayonnaise mixture on one side of 2 of the rectangles. On the other 2, arrange the chicken slices, spinach, and onions. Top with provolone cheese. Place the remaining 2 slices on top, mayonnaise side down.
Lightly grease a cast iron skillet with olive oil. Heat over medium-high heat. Add sandwiches. Cook until golden brown on the bottom, pressing down on them with a spatula. Flip over and cook until golden on the second side.
Cut each sandwich in half and serve.
