Heat the oven: Set an oven rack in the middle position and heat the oven to 425°F. Line a large rimmed sheet pan with parchment paper for easier cleanup, or leave it unlined if you want the gnocchi to brown a little more.

Dry the chicken: Put the chicken thighs on a plate and pat them dry with paper towels. Dry chicken browns better than wet chicken.

Season the vegetables and gnocchi: In a large bowl, stir together 2 tablespoons of the olive oil, the chopped garlic, the lemon juice from half the lemon, 3/4 teaspoon of the salt, and 1/4 teaspoon of the black pepper. Add the gnocchi, broccoli, and red onion, then toss until everything is lightly coated.

Season the chicken: Put the chicken thighs on the sheet pan. Drizzle with the remaining 1 tablespoon olive oil and sprinkle with the remaining 1/2 teaspoon salt and 1/4 teaspoon black pepper. Turn the thighs so both sides are seasoned, then spread them out on one side of the pan so they are not touching.

Start the chicken: Roast the chicken for 10 minutes. This gives it a head start so it can brown and release some juices before the gnocchi and vegetables go onto the pan.

Add the vegetables and gnocchi: Carefully take the sheet pan out of the oven. Spoon the gnocchi, broccoli, and onion onto the open side of the pan in a single layer, and scrape any garlic and oil left in the bowl over the vegetables.

Roast again: Return the pan to the oven and roast for 10 minutes. The broccoli should start to soften and the gnocchi should begin to look lightly golden on the bottoms.

Turn everything: Take the pan out again. Use a thin spatula or spoon to turn the gnocchi and broccoli so new sides are touching the pan, and flip the onion wedges. Turn the chicken thighs over as well so they brown evenly.

Finish roasting: Return the pan to the oven and roast for 8 to 12 minutes more. The chicken is done when it is browned and the thickest part is no longer pink in the center, the broccoli is tender at the stems, and the gnocchi has crisp golden spots.

Add the Parmesan: Sprinkle the grated Parmesan over the gnocchi, broccoli, and chicken. If any area looks dry, drizzle on a little more olive oil.

Warm the finish: Put the pan back in the oven for 1 to 2 minutes, just until the Parmesan melts a little and clings to the hot food.

Serve: Let the pan stand for 2 minutes. Serve the chicken with the crispy gnocchi, broccoli, and onion, with the lemon wedges alongside for squeezing over the top and extra Parmesan if you like.