Heat oven to 350°F and arrange a rack in center of oven. Whisk together eggs, milk, cream, and a pinch of salt in a medium bowl. Set aside.
Heat 1 tablespoon olive oil in a 10-inch cast-iron skillet over medium-high. When warm and almost smoking, add mushrooms in a single layer (you may need to do this in 2 batches). Sprinkle a pinch of salt over mushrooms and cook without disturbing for 2 minutes. Turn mushrooms and repeat, then reduce heat to medium and cook, stirring, until mushrooms have collapsed and are tender. Repeat with second batch if needed. Reduce heat to low.
Pour egg mixture over mushrooms. Scatter cheese and chives over eggs, then season with salt and pepper. Place skillet in oven and bake for about 20 minutes, until frittata is set.
Set broiler to high and place frittata under it for about 2 minutes, until the top puffs, bubbles, and turns golden brown in spots.
Let cool for 2 to 3 minutes before serving.