Stir together the mayonnaise, basil, and pepper in a cup. Spread the seasoned mayo evenly on one side of four slices of the bread.
Use a knife to spread the blue cheese on the other four slices of bread. Then layer on the strawberries, turkey, and spinach. Top with the mayo-covered slices of bread, compress the sandwiches slightly, and generously coat them all over with cooking spray.
Heat a grill pan or panini press to medium. If using a grill pan, place a heavy weight, such as a cast-iron pan or foil-wrapped brick, on the sandwiches to compress them. If using a panini press, close the lid. Cook until the bread is nicely browned and the cheese begins to soften, 3 to 5 minutes per side.
Cut each sandwich in half diagonally and serve hot.
Tip: To save time, buy crumbled blue cheese.