For the cupcakes: Preheat the oven to 400 degrees F. Line one 12-cup standard muffin tin with paper cupcake liners. Spray the liners with baking spray.
Cut a 2-inch round from each slice of bread and toast in the oven on a baking sheet until lightly browned. Save the rest for breadcrumbs. Place a piece of toast in the bottom of each cup and set aside.
Cook the ground beef in a 10-inch skillet over medium heat until browned, stirring occasionally, about 6 minutes. Stir in 1/2 teaspoon salt and the pepper. Drain off any fat, if necessary, and place the beef in a medium bowl; set aside.
Wipe out the skillet with a paper towel and cook the bacon over medium heat until crisp, turning once, about 6 minutes. Remove and drain on paper towels.
Remove all but 1 tablespoon of the bacon fat from the skillet and add the onions. Cook over medium heat, stirring frequently, until the onions are very soft, about 8 minutes. Stir in the mustard and relish. Stir into the bowl with the ground beef. Divide evenly between the prepared cups.
Place a cheese cube in each cup and push gently down into the meat mixture.
Whisk together the flour, baking powder and remaining 1/4 teaspoon salt in a medium bowl. Whisk together the milk, sour cream and eggs in a 1-quart glass measuring cup. Add the milk mixture to the flour mixture and whisk just until blended. Return to the measuring cup. Pour over the beef mixture.
Bake until a toothpick inserted into the center of the cupcakes comes out clean, 18 to 20 minutes. Cool the cupcakes in the tin on a rack for 10 minutes.
For the sauce: Combine the mayonnaise, ketchup and relish.
To assemble: Drizzle the cooled cupcakes with the sauce. Garnish with the reserved bacon and with lettuce, tomato, onions and pickles.