Step 1: stir cream cheese, shredded cheese, heavy cream, and 1/4 tsp salt together in a small bowl until well combined; set aside.
Step 2: stir mayonnaise, garlic, and 1/4 tsp salt together in a separate bowl; set aside.
Step 3: place two slices of cheddar cheese on each of 4 pieces of bread. Spread cream cheese mixture evenly among the 4 cheddar-topped slices; top each with 2 slices provolone cheese. Add remaining bread slices on top.
Step 4: spread half of mayonnaise mixture evenly over top of sandwiches.
Step 5: heat a large non-stick skillet over medium heat. Place sandwiches in skillet, mayonnaise side down. Spread the top of each sandwich with remaining mayonnaise mixture.
Step 6: cook sandwiches until golden brown on both sides, flipping occasionally until cheese is melted and bread is well toasted.