Place several paper towels over a cloth dish towel. Lay tomatoes in a single layer and sprinkle with salt. Let stand 30 minutes. Cover with more paper towels and press firmly to remove some of the moisture from tomatoes.
Heat oven to 425°.
Sprinkle ½ cup of cheddar cheese over the bottom of the piecrust. Place a layer of tomatoes over it, overlapping some, and sprinkle with ½ tablespoon of flour. Sprinkle with half of each of the cheeses, then spread ¼ cup mayonnaise as well as you can over the tomatoes. Sprinkle with basil. Layer again with tomatoes, the remaining flour and half the remaining cheese. Spread remaining mayonnaise over the top, then finish with remaining cheese.
Bake pie until crust is golden and cheese is golden brown, 35 to 40 minutes. Let pie cool at least 1 hour and up to 3 hours before slicing and serving. Serves 8. Biscuit crust for deep dish pie
Mix 1¾ cups all-purpose flour, 1½ teaspoons baking powder, ½ teaspoon baking soda and ½ teaspoon salt in a medium-sized bowl.
Cut 6 tablespoons butter into 6 pieces and drop them into flour. Use a pastry blender or your fingers to work the butter into pea-size pieces throughout the flour. Add ¾ cup buttermilk and stir just until the mixture holds together.
Wrap dough in plastic wrap and chill for an hour. Roll out dough between 2 sheets of plastic wrap to fit a deep dish piecrust, remove top layer of plastic and invert dough over pie plate. Press into pan and peel off plastic wrap.