In a large skillet, heat 1 tablespoon of olive oil over medium-high heat. Add the onion; cook until lightly browned on the edges (about 6 minutes). Transfer onion to a large bowl and set aside to cool completely.
Line 2 baking sheets with paper towels. Place sliced tomatoes in a single layer on the paper towels. Sprinkle with 1 teaspoon salt. Allow to sit for 30 minutes to remove some of the moisture, then pat dry with additional paper towels.
Once the onion is cool, stir in Swiss cheese, 1 cup of the cheddar, mayonnaise, basil, pepper, egg and ½ teaspoon salt.
Spread half of the mayonnaise mixture in the bottom of the prepared crust. Sprinkle with 1 tablespoon bread crumbs. Arrange half of the tomatoes on top. Repeat with another layer of the mayonnaise mixture, another layer of breadcrumbs, and another layer of tomatoes. Sprinkle remaining ½ cup of cheddar cheese over top.
Bake in a 350°F oven for about 45 minutes, or until brown and bubbly.
Cut into slices and serve warm or at room temperature. I find that it's easiest to slice when I allow the pie to rest for at least 20 minutes being cutting into it.