Prepare the oven and noodles. Preheat the oven to 375°F. Lightly grease a 9×13-inch baking dish. If using regular lasagna noodles, boil according to package directions until al dente, then drain and lay separately on a lightly oiled baking sheet. If using oven-ready noodles, skip boiling and check the package directions for any additional liquid needed in the sauce.
Cook the beef. Heat 1 tablespoon olive oil in a large skillet over medium heat. Add the onion and cook for 4–5 minutes, until softened. Add the beef and cook, breaking it apart, until browned and cooked through to 160°F. Stir in the garlic, ½ teaspoon Italian seasoning, salt, and pepper during the final minute. Drain excess grease if needed and transfer the mixture to a bowl.
Soften the vegetables. In the skillet, heat the remaining tablespoon of olive oil. Add the carrot and cook for 3–4 minutes. Add the zucchini and frozen cauliflower. Cover and cook for about 8–12 minutes, stirring occasionally, until the vegetables are very tender. Add a small splash of water only if needed to prevent sticking.
Remove excess moisture. Uncover the skillet and cook until the excess liquid evaporates. Add the thawed, well-squeezed spinach and the remaining ½ teaspoon Italian seasoning. Cook for another 1–2 minutes.
Blend the sauce. Add the cooked vegetables, cannellini beans, and about 1 cup of marinara to a blender. Blend until smooth, adding a little more marinara from the jar if needed to help it blend. Follow your blender’s directions for warm ingredients and work in batches if necessary. Alternatively, use an immersion blender in a deep saucepan.Pour the blended mixture into a saucepan and stir in the remaining marinara from the jar. If the sauce is too thick or you prefer more tomato flavor and a redder color, stir in extra marinara, ¼ cup at a time. Add water a tablespoon or two at a time if it still needs loosening. Taste and adjust the seasoning.The sauce should spread easily rather than hold its shape like a thick purée. For oven-ready noodles, keep it a little looser and follow the package directions for additional liquid. If you already blended in the entire jar, simply stir in extra marinara afterward as needed. Build the first layer. Spread about ¾ cup of the blended sauce over the bottom of the prepared baking dish. Arrange enough noodles to cover the base, trimming to fit if necessary. Add half the beef, half the remaining sauce, 1½ cups mozzarella-provolone, and ¼ cup Parmesan.
Build the remaining layers. Add another layer of noodles, beef, sauce, mozzarella-provolone, and Parmesan. If you have enough ingredients for more than two layers, continue layering, reserving enough sauce and cheese for the top. Finish with a layer of noodles completely covered with sauce and cheese, making sure no noodle edges are exposed.
Bake. Cover with foil, tenting it away from the cheese. Bake for 35–40 minutes. Uncover and bake for another 10–15 minutes, until bubbling and the center reaches 165°F.
Rest and serve. Let the lasagna rest for 15 minutes, then cut into 8 portions.