Preheat the oven to 450 degrees F.
Heat the butter in a large heavy skillet over medium-high heat until bubbly, then add the onion, carrot, celery, garlic, and season with 1 1/2 teaspoons kosher salt and 3/4 teaspoon pepper. Saute, stirring occasionally until vegetables are golden, about 10 minutes. Let cool.
Meanwhile, soak the breadcrumbs in milk until soft, about 5 minutes. Squeeze the bread to remove the excess milk and discard the milk.
Lightly beat the egg in a large bowl, then add in the cooled onion mixture, breadcrumbs and meatloaf mix. Gently mix until just combined. Form into 4 (4 1/2-inch) patties making a small indentation in the center of each.
Line a baking sheet with foil. Brush 1 tablespoon chili sauce on top of each patty, then place burgers on the prepared pan. Wrap each patty with 2 strips bacon, tucking bacon edges under the patties. Bake the burgers until an instant-read thermometer inserted sideways into the burger registers 160 degrees F, 14 to 16 minutes. Alternately, preheat the broiler, remove the burgers from pan and broil about 4 inches from the heat source until the bacon browns, 2 to 3 minutes.
Place the burgers on the rolls and top with chili sauce and lettuce leaves, if desired.