Preheat oven to 350 degrees F (175 degrees C). Grease a 9-inch pie dish.
Heat olive oil in a skillet over medium heat. Add onion; cook and stir for 2 minutes. Add thyme, leeks, and mushrooms; cook and stir until the onions are cooked through, 5 to 8 minutes.
Add sweet potato; cook and stir until the sweet potato is tender, about 15 minutes. Place mixture in prepared pie dish.
Combine eggs, sour cream, salt, and cream in a food processor; pulse until smooth. Pour into pie dish and stir slightly to distribute over onion mixture. Top with Asiago cheese.
Bake in preheated oven until top is beginning to turn golden brown and center is gently firm, about 45-50 minutes. Let sit a few minutes before cutting. Serve warm.
