Prep the grill or pan. Preheat a gas or charcoal grill to medium-high heat (about 400–425°F). If using a skillet or grill pan, heat over medium-high and lightly oil the surface.
Mix the burger base. In a bowl, combine ground turkey, grated onion, mayo, Dijon, Worcestershire, smoked paprika, garlic powder, salt, and pepper.
Stir gently with a fork until just combined—don’t overmix.
Form the patties. Divide the mixture into 4 equal parts. Shape into patties about 1/2 to 3/4 inch thick. Use your thumb to press a small dimple in the center of each patty to prevent puffing.
Oil and season the surface. Lightly brush patties with olive oil.
Sprinkle a pinch more salt and pepper on top for a crisp, seasoned crust.
Grill the patties. Place patties on the hot grill or pan. Cook for 4–5 minutes per side, flipping once, until the internal temperature reaches 165°F. If adding cheese, place it on during the last minute to melt.
Toast the buns. Split buns and toast cut-side down on the grill for 30–60 seconds, until lightly golden.
Steam the broccoli. While burgers rest for a few minutes, bring 1 inch of water to a simmer in a pot.
Add a steamer basket with broccoli florets, cover, and steam for 4–5 minutes until bright green and just tender. Drain well.
Finish the broccoli. Toss florets with olive oil or butter, a squeeze of lemon, salt, and pepper. Add red pepper flakes or Parmesan if you like.
Assemble and serve. Spread a little Dijon or mayo on buns.
Layer lettuce, turkey patty, tomato, and onion. Serve with the steamed broccoli on the side.