Preheat oven to 350°. Set a large skillet over mediump-high heat and add butter. Add zucchini and 1/2 teaspoon salt; cook, stirring often, until just translucent, about 5 minutes. Use a slotted spoon to remove zucchini, and set aside.
Add oil to pan and return to medium-high heat; add onion and garlic. Cook, stirring often, until onions are lightly golden, about 10 minutes. Remove from heat.
In a large bowl, stir together eggs, cottage cheese, Parmesan, spinach, parsley, onion/garlic mixture, remaining 1 teaspoon salt, and pepper.
In 9x13-inch pan, layer ingredients like lasagna: a layer of tomato sauce, a layer of zucchini, a light sprinkling of breadcrumbs, a layer of cheese/spinach mixture, and a layer of mozzarella. Repeat, ending with a layer of sauce and a sprinkling of Parmesan.
Cover dish with foil and bake 30 minutes. Remove foil and bake until top is browned and bubbling, about 30 minutes more.