Heat the 1/4 cup olive oil in a large skillet over medium-high heat. Add the garlic; cook, stirring, until golden, 1 minute. Add the tomatoes, 1 tablespoon of the herbs, 1/2 teaspoon salt, and pepper to taste; cook until saucy, about 5 minutes. Transfer to a blender and carefully blend until smooth (see Cook's Note). Return 1 cup of the sauce to the skillet and reduce the heat to low; reserve the remaining sauce.
Meanwhile, combine the ricotta, egg, the 2 tablespoons Parmesan, the remaining 3 tablespoons herbs, 1/2 teaspoon salt, and pepper to taste in a medium bowl.
Place 2 lasagna noodles over the sauce in the skillet. Layer half of the carrots and zucchini on top; drizzle with olive oil and season with salt and pepper. Cover with half of the spinach, half of the ricotta mixture, a few pieces of mozzarella and 2 to 3 tablespoons of the reserved tomato sauce. Repeat the layers, ending with noodles. Top with the remaining sauce and mozzarella. Cover and simmer until the lasagna is cooked and the mozzarella melts, 20 to 25 minutes.
Let rest for a few minutes before slicing. Garnish with more Parmesan and fresh herbs.
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.