In a small bowl, stir together 1/4 cup mayonnaise and the minced garlic. Cover and refrigerate until assembly.
Preheat oven to 375°F.
While the oven preheats, heat 1 tablespoon olive oil in a large nonstick skillet over medium heat. Add the thinly sliced red onion and sauté 3 minutes. Add 2 teaspoons granulated sugar, 1 tablespoon balsamic vinegar, and 1/4 teaspoon crushed red pepper flakes; continue sautéing about 5 minutes, until the onions are softened. Transfer the onions to a bowl and wipe the skillet clean.
Unroll the 10-ounce refrigerated pizza crust into a 15×10-inch jellyroll pan and press to fit the pan. Bake in the preheated oven for 10 minutes. Remove and let cool, then cut the baked crust into 4 equal rectangles.
While the crust cools, prepare the chicken: flatten the two boneless, skinless chicken breasts to about 1/4-inch thickness with a meat mallet or rolling pin. Season both sides with salt, pepper, and 1/4 teaspoon dried thyme.
Heat the remaining 1/2 tablespoon olive oil in the wiped-clean skillet over medium-high heat. Add the flattened chicken and cook 4 to 5 minutes per side, until cooked through. Transfer the chicken to a cutting board, let rest 5 minutes, then slice.
Make sure the 9-ounce package of frozen chopped spinach is thawed and well squeezed to remove excess liquid.
To assemble 2 sandwiches: spread the mayonnaise mixture on one side of 2 of the crust rectangles. On the other 2 rectangles, divide and layer the sliced chicken, the squeezed spinach, and the cooked onions. Divide the 8 provolone slices evenly and place them on top of the fillings. Place the mayonnaise-spread rectangles on top of each sandwich, mayonnaise side down.
Lightly grease a cast-iron skillet with a thin coating of olive oil (use any oil remaining in the chicken skillet if available). Heat the skillet over medium-high. Add the sandwiches and cook until the bottoms are golden brown, pressing down gently with a spatula. Flip the sandwiches and cook until the second side is golden brown.
Remove sandwiches from the skillet, cut each sandwich in half, and serve.