Sunny Side Smash Burger with Salsa Verde

Heat broiler. Toss tomatillos, onion, and jalapeños with oil on a foil-lined rimmed baking sheet; season with salt. Broil until blackened in spots and very soft, 12–15 minutes. Let vegetables cool slightly, then transfer to a food processor. Add cilantro and black pepper; pulse until coarsely chopped (3 or 4 pulses). Taste salsa verde and season with more salt if needed.

Portion beef into 4 loosely packed balls (5 oz. each). Heat a large cast-iron skillet over medium-high until just beginning to smoke. Add 2 Tbsp. oil, then all 4 balls of meat. Firmly smash down on meat with a heavy metal spatula to create a craggy-edged 4"-diameter patty; season with salt and pepper. Cook, undisturbed, until deeply charred underneath, about 3 minutes. Flip and season other side with salt and pepper; cook to medium, about 3 minutes. Transfer to a plate and let rest.

Carefully wipe out skillet with a paper towel, leaving only a light coating of oil. Set skillet over medium heat. Working in 2 batches, toast buns, cut side down, until golden brown, about 1 minute. Transfer to a clean surface.

Heat remaining 2 Tbsp. oil in a large nonstick skillet over medium. Crack eggs into skillet and cook until whites are just set and edges are crisp, about 4 minutes. Season with salt.

Spread insides of both tops and bottoms of buns with a heaping tablespoonful of salsa verde. Place a burger on each bottom bun, top with an egg, sunny side up, and close burger.

Salsa verde can be made 1 day ahead. Transfer to a small bowl; cover and chill.