Pulse mushrooms and garlic in a food processor until finely chopped.
Heat olive oil in a 12-inch heavy skillet over medium-high heat until it shimmers. Season chicken with 1/2 teaspoon salt and 1/4 teaspoon pepper and cook, stirring occasionally, until just golden, about 3 minutes. Transfer with a slotted spoon to a bowl.
Reduce heat to medium and cook onion, stirring occasionally, until softened, about 3 minutes.
Add mushroom mixture, rosemary, 1/2 teaspoon salt, and 1/4 teaspoon pepper and cook, stirring constantly, until beginning to brown, about 4 minutes.
Add balsamic vinegar and cook until evaporated. Add chicken and tomatoes (with juice), then simmer, breaking up tomatoes with a spoon and stirring occasionally, until sauce is just thickened, about 15 minutes.
Meanwhile, cook pasta in a pot of boiling salted water (2 tablespoons salt for 6 quarts water) until al dente.
Add arugula to sauce and stir until wilted. Stir in drained pasta and cook 1 minute. Serve with Grated Parmigiano-Reggiano and enjoy.
