Step 1

Red Sauce: Heat a straight-sided skillet over medium. Swirl in oil; add onion and 1 teaspoon salt. Cook, stirring occasionally, until onion is soft and translucent, 7 to 9 minutes. Stir in garlic and red-pepper flakes; cook 30 seconds. Transfer one-third of onion mixture to a bowl and reserve for meatballs. Add tomatoes to skillet; bring to a boil. Reduce heat and simmer, stirring occasionally, until reduced and thickened slightly, about 20 minutes.

Step 2

Meatballs: While sauce simmers, heat a large skillet over medium-high. Swirl in oil. Add mushrooms and cook, stirring occasionally, until soft and most of moisture has evaporated, 8 to 10 minutes. Transfer to bowl with reserved onion mixture; let cool slightly.

Step 3

Preheat broiler with rack 4 inches below heating element. Stir breadcrumbs and milk into mushroom mixture until milk is absorbed. Add cheese, parsley, oregano, egg, 1 teaspoon salt, and 1/4 teaspoon pepper. Add beef and gently fold with your hands until evenly combined (don't overwork, or meatballs will be tough). Use an ice cream scoop, or roll heaping tablespoons between dampened palms, to form thirty 1 1/2-inch balls. Transfer to a rimmed baking sheet brushed with oil. Broil, undisturbed, until browned on top, about 5 minutes.

Step 4

Transfer meatballs to skillet with red sauce. Simmer, stirring gently, until just cooked through, about 5 minutes. Meanwhile, boil pasta in a large pot of generously salted water until al dente, about 2 minutes less than per package instructions. Drain pasta and return to pot; toss with 1 1/2 cups sauce. Serve, topped with meatballs, more sauce, cheese, red-pepper flakes, and parsley.