Step 1

In a small bowl, stir together the ricotta, Parmesan, pepper flakes, oregano and ¼ teaspoon each salt and pepper; set aside. In a broiler-safe 12-inch skillet over medium-high, heat the oil until shimmering. Add the sausage, garlic, ½ teaspoon salt and ¼ teaspoon pepper; cook, breaking the sausage into small bits, until browned, 5 to 8 minutes.

Step 2

Add the tomatoes with juices, the tomato paste and 1½ cups water. Bring to a simmer, then reduce to medium and cook, uncovered and stirring, until slightly thickened and dots of oil have formed on the surface, 12 to 15 minutes. A few at a time, insert the broken lasagna noodles in the sauce, standing them up on a frilly edge; the pieces will be only partially submerged. Cover and cook, undisturbed, until the noodles begin to soften and slump into the sauce, 6 to 8 minutes.

Step 3

Uncover, add another ½ cup water and stir, fully coating the noodles and submerging them in the sauce. Re-cover and cook, using a silicone spatula to scrape along the bottom of the pan and stir about every 5 minutes, until the pasta is al dente, 15 to 20 minutes. Near the end of cooking, if the sauce has thickened and the noodles begin to stick to the pan before they are al dente, stir in another ¼ cup water. Remove from the heat and let stand, covered, for 5 minutes. Meanwhile, heat the broiler with a rack about 6 inches from the element.

Step 4

Uncover the skillet. Dollop spoonfuls of the ricotta mixture onto the surface, then sprinkle evenly with the mozzarella. Broil, uncovered, until the surface and edges are brown, 2 to 3 minutes.