Combine bacon and onion in a large skillet and cook over medium-high heat, turning occasionally, until evenly browned, about 10 minutes. Set aside.

Heat oil in a pan over medium heat. Add chicken and 1/2 of the lemon juice and cook until no longer pink in the centers and juices run clear, 5 to 7 minutes per side.

Meanwhile, mix balsamic vinegar, brown sugar, rosemary, red pepper flakes, and remaining lemon juice together in a saucepan and bring to a boil. Reduce heat to medium-low and simmer until mixture bubbles, 5 to 7 minutes. Add tomatoes with some of their juice and continue to simmer while chicken finishes cooking.

Melt butter in a pan over medium heat. Pan-fry gnocchi in the hot butter until golden brown, 5 to 7 minutes.

Place a steamer insert into a saucepan and fill with water to just below the bottom of the steamer. Bring water to a boil. Add spinach, cover, and steam until tender, 2 to 6 minutes.

Place bacon and onion onto a serving plate and top with spinach, gnocchi, and chicken, in that order. Drizzle with tomato-balsamic sauce.