Heat a grill, grill pan or panini press to medium heat. Split the focaccia in half through the side to make two large rounds.
Spread the mayonnaise and mustard over the top round of the bread. Arrange the provolone over the bottom round of bread. Layer the turkey, tomato, and spinach over the cheese. Top with the spread round of bread and coat sandwich all over with cooking spray.
Grill the entire round sandwich over medium heat. Put a heavy weight, such as a cast-iron pan or foil-wrapped brick, over the sandwich to compress it. Or, if using a panini press, press down the lid. Cook until nicely marked by the grill and the cheese melts, 3 to 5 minutes per side, turning 90 degrees halfway between cooking on each side to make crosshatched marks. Cut into 4 wedges and serve.
Tip: To save time, replace the mayonnaise and mustard with store-bought Dijonnaise.