Preheat the oven to 400°F (200°C). Place the sausages, peppers, and mushrooms on a rimmed baking sheet. Drizzle with 2 tablespoons of the olive oil, then season with 1/2 teaspoon salt and 1/4 teaspoon pepper.
Bake, tossing the vegetables and turning the sausages a few times during cooking, until the sausages are browned and the vegetables are tender, about 30 minutes.
In a large bowl, toss the kale with the remaining 1 tablespoon olive oil and 1/4 teaspoon salt. Add the kale to the pan with sausage and vegetables, toss gently together, and cook until the kale has wilted, about 5 minutes more. Serve.