Preheat oven to 425 degrees.
Mix eggs with milk, salt and pepper in a medium bowl.
Heat one tablespoon olive oil over medium-high heat in a 10-inch cast iron skillet or other oven-safe skillet.
Sauté the onion until is begins to soften, 2-3 minutes. Add the mushrooms and sauté, stirring frequently, until golden brown and caramelized, about 8 minutes more. Cooking the mushrooms really well brings out their best flavor.
Add the spinach and cook until completely wilted, another 1-2 minutes. Add another tablespoon of olive oil if the skillet become dry at any point.
Reduce the heat to medium-low and pour in the egg mixture. Cook, stirring gently, until the eggs begin to set but do not begin to scramble.
At this point, gently stir in 1/2 cup of Fontina and cook for 2-3 minutes more, without stirring, until eggs begin to set around the edges.
Transfer the pan to the oven and bake until the eggs are just set in the middle, 4-5 minutes.
Remove from oven and switch to broil setting.
Sprinkle the remaining 1/4 cup cheese over the top and broil for 1-2 minutes, watching very closely, or until top is golden brown.