A cherry-tomato slice and a pickle coin are sandwiched, along with the burger, between halves of a fresh poppy-seed brioche.
Preheat convection oven to 375 degrees or conventional oven to 350 degrees. Coat parchment-lined baking sheet with cooking spray. Roll dough into 3/4-inch balls; place 1 1/2 inches apart on baking sheet. Cover with plastic wrap sprayed lightly with cooking spray. Let stand in a warm place until dough doubles in bulk, 20 minutes.
Brush egg wash over dough; sprinkle with poppy seeds. Bake until golden brown, 8 to 10 minutes, rotating pan halfway through. Cool buns on rack.
In bowl, mix ground beef, salt, and pepper. Shape 1/4 ounce at a time into 1 1/2-inch patties. Broil or grill 1 minute per side.
At serving time, preheat oven to 350 degrees. Slice buns in half. Layer ketchup, tomato, and lettuce on each bottom half.
Place burgers on baking sheet; top each with a slice of cheese. Bake until cheese melts, 2 to 3 minutes. Place in buns; serve.