Preheat the oven to 375°F (190°C).
In a skillet, heat olive oil over medium heat. Add the diced potatoes and turnips. Cook until tender and slightly golden, about 8-10 minutes.
(Optional) Add diced onions and garlic to the skillet and sauté until softened, about 2-3 minutes.
In a mixing bowl, whisk together eggs, milk (or cream), salt, and pepper.
Pour the egg mixture over the cooked vegetables in the skillet. Stir gently to combine.
Sprinkle shredded cheese and fresh herbs (if using) over the top of the frittata.
Transfer the skillet to the preheated oven and bake for 15-20 minutes, or until the eggs are set and the top is slightly golden.
Remove from the oven, let it cool for a few minutes, then slice and serve.