Heat the oven to 375°F with a rack in the middle; have a 10-inch oven-safe nonstick or cast-iron skillet ready.
Prep the produce: slice the mushrooms, finely chop the shallot, separate the Swiss chard stems from leaves (chop stems and coarsely chop leaves), and mince the garlic; finely grate the Parmigiano-Reggiano.
Warm the olive oil and butter in the skillet over medium-high heat until the butter foams. Add the mushrooms and cook, stirring only occasionally, until browned and most liquid evaporates, 6–8 minutes.
Add the shallot and the chopped Swiss chard stems. Cook, stirring, until the stems soften and the shallot turns translucent, 3–4 minutes.
Stir in the garlic and thyme and cook until fragrant, 30–60 seconds.
Add the Swiss chard leaves and cook, stirring, until wilted and any moisture cooks off; spread the vegetables in an even layer and remove the pan from the heat, 2–3 minutes.
In a large bowl, whisk the eggs with the whole milk, all but 2 tbsp of the Parmigiano-Reggiano, the kosher salt, and the black pepper until smooth and slightly frothy, 30–45 seconds.
Return the skillet to medium heat and pour in the egg mixture. Using a spatula, gently stir and scrape around the edges for 1–2 minutes until the edges just begin to set; smooth the top and sprinkle the remaining cheese over the surface.
Transfer the skillet to the oven and bake until the frittata is puffed around the edges and the center is just set with a slight jiggle, 8–12 minutes (a knife inserted near the center should come out clean).
Let rest 5 minutes, run a spatula around the rim to loosen, then slice into wedges and serve warm or at room temperature.