1Warm 2 Tbsp. extra-virgin olive oil in a large Dutch oven or other heavy large pot over medium-high heat. Cook 1 lb. spicy Italian sausage, casings removed, 1 medium onion, finely chopped, 3 large garlic cloves, chopped, 2 tsp. dried oregano, and ¼ tsp. crushed red pepper flakes, stirring often and breaking up sausage into small pieces, until sausage is cooked through, about 10 minutes. Add one 28-oz. can crushed tomatoes with added purée and one 14.5-oz. can diced tomatoes with green peppers and onions. Bring to boil, then reduce heat and simmer, stirring occasionally, until flavors are melded, about 5 minutes. Season with kosher salt and freshly ground black pepper. Do Ahead: Sauce can be made 1 day ahead. Let cool; cover and chill.
2Pulse 1½ cups (packed) fresh basil leaves in a food processor until finely chopped. Add 1 large egg, one 15-oz. container plus 1 cup part-skim ricotta cheese, 6 oz. low-moisture mozzarella cheese, coarsely grated (about 1½ cups), 2 oz. Parmesan cheese, coarsely grated (about ¾ cup), ½ tsp. Diamond Crystal or ¼ tsp. Morton kosher salt, and ¼ tsp. freshly ground black pepper. Pulse until combined but still a little chunky.
3Place a rack in middle of oven; preheat oven to 375°. Spread 1¼ cups of meat sauce in a 13x9" glass baking dish. Arrange 3 no-boil lasagna noodles in a single layer over sauce. Spoon 1½ cups ricotta mixture over the layer of noodles and spread it from edge to edge. Scatter ¾ cup coarsely grated low-moisture mozzarella cheese and ¼ cup coarsely grated Parmesan cheese evenly over. Starting with another layer of sauce, repeat process 2 more times. Arrange remaining 3 noodles on top; spread remaining sauce over the noodles. Scatter remaining mozzarella and Parmesan evenly over.
