Step 1Whisk together the eggs, one-fourth cup creme fraiche and the cheese until thoroughly combined. Set aside.
Step 2Heat a large nonstick skillet or cast iron pan over medium-high heat. Add the olive oil and butter and heat until the butter bubbles but is not yet turning brown. Add the shallots and garlic and cook over medium to medium-low heat until tender and beginning to brown, about 3 to 4 minutes.
Step 3Add the mushrooms and thyme and cook until the mushrooms are tender, about 2 minutes. Season to taste with salt and pepper.
Step 4Stir the egg mixture to recombine, then pour it over the mushroom mixture. Reduce the heat to medium-low. Use a rubber spatula to even out and distribute the mushroom-shallot mixture. Cook until the bottom is set and the edges begin to shrink a little, about 5 to 8 minutes. Meanwhile, set the oven rack at least 6 inches away from the broiler and heat the broiler.
Step 5When the bottom is set, finish the frittata by broiling briefly until the top is just set, 1 to 2 minutes. Do not brown.
Step 6Remove the frittata from the oven and loosen it from the pan with a spatula. To serve, cut into wedges and top each with a dollop of creme fraiche and a sprinkling of chopped chives and sea salt.