Preheat the oven to 400°F (200°C). Line a 9-by-9-inch pan with aluminum foil and spray with cooking spray; set aside.
In a large mixing bowl, combine the grated zucchini, grated carrot, salt, and black pepper, then pour the beaten egg over the vegetables and toss to coat evenly.
Press the vegetable mixture into the prepared pan, smoothing the top with a spatula to form an even crust.
Bake the crust for 25 to 28 minutes, or until it begins to brown lightly and has firmed up but is not watery.
Remove the crust from the oven and spread the sour cream evenly over the top in a smooth layer.
Sprinkle about 2 cups of the shredded cheese evenly over the sour cream layer.
Drizzle the 16-ounce jar of salsa evenly over the cheese layer.
Top with the remaining approximately 1½ cups shredded cheese, distributing it uniformly.
Bake for about 35 to 40 minutes, or until the top is bubbly and golden brown to your liking.
Let the lasagna cool for at least 10 minutes before slicing and serving.