This egg recipe, courtesy of chef Pierre Schaedelin, earned its name by becoming a favorite of Martha's friend and editorial director of decorating, Kevin Sharkey.
Preheat oven to 325 degrees. Using a small spoon, carefully remove seeds from tomatoes and discard. Place tomatoes, cut side-down, on a small parchment paper-lined baking sheet along with garlic and thyme. Drizzle with olive oil and season with salt and pepper. Transfer to oven and bake until softened, 45 to 50 minutes. Remove from oven and set aside.
Melt 2 tablespoons butter in a medium skillet over medium-low heat. Add onions and cook, stirring occasionally, until caramelized, about 30 minutes. If onions begin to brown too quickly, add 3 to 4 tablespoons water to prevent burning and sticking. Remove from heat and set aside.
Preheat the oven to broil. Heat 3/4 teaspoon butter in a small nonstick skillet over medium heat. Carefully crack 2 eggs into skillet, side by side. Cook eggs until whites are just set, 2 to 3 minutes. Place a tomato half and 2 tablespoons caramelized onions on the whites in between the two yolks. Carefully top eggs, tomatoes, and onions with 2 to 3 slices of cheese to cover. Transfer to broiler just until cheese melts, about 15 seconds.
Gently transfer eggs onto a plate and season with salt and pepper. Sprinkle with chopped basil and serve immediately. Repeat process with remaining eggs, tomatoes, onions, and cheese.