Stir the pesto and mayonnaise together until smooth.
Warm a large skillet over medium heat. Add the olive oil and butter.
Toast the cut sides of the bread in the skillet until lightly golden.
Spread the pesto mayo over the toasted sides of the bread.
Layer turkey, provolone, tomato, and spinach on half the bread. Season lightly.
Close sandwiches and return to the skillet. Press and cook until cheese melts.
Drizzle with balsamic glaze if using, slice, and serve warm.