Heat olive oil in large skillet over medium heat.
Melt butter in saucepan for béchamel sauce preparation.
Whisk flour into melted butter until smooth paste forms.
Gradually add milk while whisking to prevent lumps forming.
Cook sauce until thickened, about 5 minutes total time.
Season béchamel with salt, pepper, and Parmesan cheese.
Layer prosciutto and mozzarella on four bread slices.
Sprinkle fresh basil over the cheese and meat layers.
Top with remaining bread slices to complete sandwiches.
Spread béchamel sauce generously on sandwich tops and sides.
Place sandwiches in heated skillet, cooking 3-4 minutes.
Flip carefully and cook another 3-4 minutes until golden.
Serve immediately while cheese is still melting hot.