Preheat the oven to 300 degrees F.
Season the tomato slices on a baking sheet with 1 teaspoon of the olive oil and some salt and pepper. Place a basil leaf on top of each slice and bake until the tomatoes are wilted, about 10 minutes.
Heat a large nonstick pan, add the remaining 1 teaspoon olive oil and place 2 ring molds in the pan. Crack the eggs into the molds, reduce the heat to low and cover. Cook over low heat until the whites have cooked through, about 4 minutes. Season with salt and pepper and top with chives.
Spread the mayonnaise on each slice of toast. Divide the bacon, lettuce and tomato between the bread slices. Place an egg on top of each sandwich and serve.