Cook and stir ground beef, Italian sausage, onion, and garlic in a large skillet over medium heat until meat is crumbly and no longer pink, about 10 minutes; drain grease. Stir in basil, oregano, brown sugar, pepper flakes, 1 teaspoon salt, and black pepper until incorporated. Stir in tomatoes and tomato paste; bring to a boil. Reduce heat to low; simmer for 30 minutes, stirring occasionally.

Preheat the oven to 375 degrees F (190 degrees C). Grease a 9x13-inch baking dish.

Combine ricotta cheese, 1 cup Parmesan cheese, eggs, parsley, and remaining 1 teaspoon salt in a large bowl. Set aside.

Remove skin slices lengthwise from zucchini, alternating with strips of remaining skin. Cut zucchini into long strips to resemble lasagna noodles. Discard seedy middle parts.

Place 1/3 zucchini strips into bottom of the prepared baking dish, filling in any gaps with scrap pieces zucchini. Spread 1/2 ricotta mixture over zucchini; top with 1 cup mozzarella, then 1/3 meat sauce. Layer in 1/2 remaining zucchini strips, remaining ricotta mixture, 1 cup mozzarella cheese, and 1/2 remaining meat sauce. Place remaining zucchini strips and remaining meat sauce on top; sprinkle with 2 tablespoons Parmesan cheese.

Bake in the preheated oven until lasagna is bubbling and top browned, about 1 hour; if top is browning too quickly, cover the dish with aluminum foil during the last 15 minutes of baking time. Remove lasagna from the oven; sprinkle remaining 1 cup mozzarella over top.

Bake in the oven until mozzarella melted, 5 to 10 minutes more. Set lasagna aside before serving for 10 minutes.