Heat the oven to 350°F and arrange a rack in the middle. Generously coat the wells of a 12-well muffin pan with butter; set aside.
Heat 1 tablespoon of the oil and the measured butter in a medium frying pan over medium heat until the oil is shimmering and the butter has melted.
Add the diced onion and cook, stirring rarely, until it is a deep golden brown, adjusting the heat as necessary if it starts to brown too much, about 30 minutes. Season well with salt and pepper, add the thyme, and stir to combine. Transfer the mixture to a medium bowl.
Meanwhile, clean, trim, and slice the mushrooms 1/2 inch thick; set aside. Crumble the goat cheese into pea-sized pieces and set aside.
Return the frying pan to medium heat, add the remaining tablespoon of oil, and heat until shimmering. Add the sliced mushrooms, season with salt and pepper, and cook, stirring rarely, until golden brown, about 10 minutes. Add to the bowl with the onion. Add the goat cheese to the bowl and stir to evenly combine; set aside.
Place the eggs and milk in a large bowl and whisk until the eggs are broken up and evenly combined with the milk, about 1 minute. Add the measured salt and whisk to combine.
Divide the onion-mushroom-cheese mixture evenly among the wells of the prepared muffin pan. Fill each well almost to the top with the egg mixture.
Bake until each frittata is puffed and the center is just set, about 12 to 15 minutes. Remove the pan to a wire rack until it’s cool enough to handle, about 5 minutes (the frittatas will deflate). Run a small knife around the perimeter of each well to loosen and remove the frittatas. Serve warm or at room temperature.
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