Preheat oven to 375 degrees F.
Slice the tomatoes, discarding the stem and root ends, into 1/4-inch slices and lay out on paper towels. Sprinkle with salt and let drain for 15 minutes. Pat try and season again with salt and pepper.
Saute onion in 1 Tablespoon each of olive oil and butter. Cook and stir over medium until starting to brown and caramelize.
Separate egg, reserving yolk. Brush the pie crust with egg white and let dry. Sprinkle about 1/3 of the bread crumbs in the bottom of the pie crust. In a small bowl combine the mayonnaise with the reserved egg yolk and stir until smooth.
Place a layer of tomatoes in the bottom of the pie crust over the breadcrumbs, using about half of the tomatoes, then top with half of the onions. Sprinkle with half of basil, half of the Manchego and half of the Parmesan cheeses. Drizzle with 1/2 of the mayonnaise-egg mixture. Repeat. Top with remaining 1/3 of breadcrumbs and drizzle with 1 T olive oil and sprinkle with the additional 2 T of Parmesan.
Bake until bubbly hot and golden brown, about 1 hour, or until it looks set. Remove from the oven and allow to cool for 15- 30 minutes before slicing and serving. Serve warm or at room temperature. Serves 6