Preheat the oven to 375°F (190°C) and grease a baking dish.
Sauté the veggies – heat olive oil in a skillet, add onions, celery, and mushrooms, and cook until softened. Add garlic, sage, cayenne, salt, and pepper.
Combine the turkey – mix the cooked turkey with the sautéed vegetables in a bowl.
Whisk the sauce – beat the eggs with heavy cream in a bowl, then stir into the turkey mixture.
Fold in the cheese – mix in shredded Gruyère and transfer everything to the baking dish.
Bake for 20–25 minutes until golden brown and bubbly.
Rest for 5 minutes, then serve warm.