1 medium onion, chopped
10 oz. portobellos, trimmed and quartered
2 tablespoons plus 2 teaspoons olive oil, divided
1 lb. ground beef
2 celery ribs
1/2 Granny Smith apple, cored (unpeeled)
1 tablespoons mayonnaise
1 1/2 teaspoons cider vinegar
1 teaspoon olive oil
1 teaspoon whole-grain mustard
Rounded 1/4 teaspoon sugar
Cook vegetables for burgers:
Pulse onion and mushrooms in a food processor until finely chopped. Heat 2 tablespoons oil in a 10-inch heavy skillet over medium heat until it shimmers. Add mushroom mixture, 3/4 teaspoon salt, and rounded 1/4 teaspoon pepper and cook, stirring occasionally, until vegetables begin to brown and liquid has evaporated, 8 to 10 minutes. Transfer to a bowl and cool to warm, stirring occasionally, about 15 minutes.
Meanwhile, make slaw:
Cut celery and apple into 2-inch-long thin julienne with slicer. Whisk together mayonnaise, vinegar, oil, mustard, sugar, rounded 1/8 teaspoon salt, and 1/8 teaspoon pepper in a medium bowl. Add celery and apple, tossing to coat. Let stand 15 minutes to develop flavors.
Finish and cook burgers:
Mix hamburger into cooled mushroom mixture with your hands until well combined. Form into 4 (4-inch) patties. Heat remaining 2 teaspoons oil in a 12-inch heavy skillet over medium-high heat until it shimmers. Cook burgers, turning once, about 8 minutes total for medium-rare.
Serve burgers, topped with slaw, on buns.