Preheat the oven to 400°F (200°C). Line a 9x9-inch baking pan with aluminum foil and spray the foil with cooking spray; set aside.
In a large mixing bowl, combine the grated zucchini, grated carrot, salt, and black pepper. Pour the beaten egg over the vegetables and toss until evenly coated.
Transfer the vegetable mixture to the prepared pan and lightly pack and smooth it into an even crust using a spatula.
Bake the crust for 25–28 minutes, until it is set and shows slight browning on top and is no longer watery.
Remove the pan from the oven and spread the sour cream evenly over the baked vegetable crust.
Sprinkle about 2 cups of the shredded cheese evenly over the sour cream.
Drizzle the jar of salsa evenly over the cheese layer.
Top with the remaining 1½ cups shredded cheese, sprinkling it uniformly over the salsa.
Return the pan to the oven and bake for about 35–40 minutes, until the top is bubbly and golden brown to your liking.
Allow the lasagna casserole to cool at least 10 minutes before slicing and serving.