Don't have mini tart pans? A muffin pan will also work -- just increase the baking time by one to two minutes.
Preheat oven to 350 degrees. Stir 4 tablespoons butter until smooth. On a clean work surface, lightly roll a bread slice with a rolling pin until it's 1/8 inch thick. Brush both sides with butter and fit into a 4-inch tart pan with a removable bottom. Use a paring knife to trim edges flush with pan, then season with salt. Repeat with remaining bread. Transfer pans to a rimmed baking sheet and bake until crisp and golden around edges, 12 to 14 minutes. Let cool completely, then remove from pans.
Whisk together eggs, milk, and 1/4 teaspoon salt. Heat a large nonstick skillet over low. Melt remaining 2 tablespoons butter, then add egg mixture and cook, stirring occasionally with a wooden spoon, until just set into soft curds, about 15 minutes.
Spoon eggs into tart shells, sprinkle with pepper, and serve, topped with bacon and sprouts.
