Arrange a rack in the middle of the oven and preheat to 350°F.
Cut a deep V into the top of each demi baguette. Remove the tops and soft interior, creating a ½-inch thick wall. Save the bread scraps for another use. Transfer the baguette boats to a large rimmed baking sheet.
In a large measuring cup or a bowl with a pour spout, whisk the eggs with the sour cream, mustard, salt, and pepper.
Divide the cooked filling ingredients between each baguette and top with half of the cheese. Divide the egg custard between each baguette boat, leaving ¼-inch of space at the top. Let sit for 10 minutes, allowing the liquid to soak into the bread. Add more egg custard, if needed. Sprinkle the remaining cheese over the top. Brush the outsides of each baguette with melted butter.
Bake the baguette boats until the egg is set and the top is beginning to brown, 30 to 35 minutes. Let cool for 5 minutes before cutting into 2-inch slices. Garnish with fresh herbs or sliced scallions and sprinkle with flaky sea salt. Serve right away.