Unroll pizza crust, and press into a 15- x 10-inch jellyroll pan.
Bake at 375° for 10 minutes. Cool completely.
Prepare spinach according to package directions. Drain well, pressing between paper towels. Set aside.
Stir together mayonnaise and half of minced garlic in a small bowl; chill.
Sauté onion in hot oil in a saucepan over medium-high heat 2 to 3 minutes or until crisp-tender. Add sugar and vinegar. Reduce heat to low; simmer, stirring occasionally, 3 to 5 minutes or until almost all liquid evaporates.
Place chicken between 2 sheets heavy-duty plastic wrap or wax paper; flatten to 1/4-inch thickness using a meat mallet or rolling pin. Sprinkle with Italian seasoning and remaining garlic.
Brown chicken in a large nonstick skillet coated with cooking spray over medium-high heat; cook 8 minutes, turning once.
Cut pizza crust into 4 rectangles. Remove rectangles from pan; spread 1 side of each with 1 tablespoon garlic mayonnaise. Top 2 rectangles evenly with chicken, spinach, onion mixture, and cheese; top with remaining 2 rectangles, mayonnaise sides down.
Cook sandwiches in a cast-iron skillet over medium-high heat. Top with a smaller skillet or press down with spatula to flatten sandwiches slightly, cooking 2 to 3 minutes on each side or until golden brown and cheese melts. Cut each sandwich into fourths.
