Prepare the pesto aioli by combining mayo, pesto and lemon juice (to taste) in a small bowl.
Spread 1 tbsp of aioli on each of the 4 bread tops.
On each bread bottom, layer the ingredients as follows:1 slice of provolone, 2 slices of tomato, salt + pepper, 6 artichoke heart halves, 2 torn basil leaves, 1/4 of the mozzarella slices. Place the bread tops on.
Preheat a grill pan or non-stick skillet over medium heat. You may also preheat a cast iron skillet, large soup pot, or foil-covered brick (any heavy cooking tool) on a separate burner to serve as a "press".
Oil the grill pan with olive oil. Place your sandwiches in the pan, working in batches if necessary. Brush or drizzle olive oil on top of the sandwiches to prevent them sticking to the bottom of your "press".
Place the "press" on top of the sandwiches and cook for 3 to 4 minutes. Carefull remove the "press" with oven mitts. Flip the sandwiches over, replace the "press" and cook an additional 3 to 4 minutes.