Heat butter in a large high-sided skillet over high heat. Add onion, garlic, thyme; season with salt and pepper. Cook until lightly caramelized, 10 to 12 minutes (adjust heat if browning too quickly). Transfer onion mixture to a plate, and let cool.
Meanwhile, whisk together eggs, milk, Parmigiano-Reggiano, and 1 teaspoon salt in a bowl. Shingle bread slices in a 10-by-14-inch oval baking dish layering with cooked onion mixture, speck, and fontina cheese. Pour over egg mixture, pressing bread gently into liquid. Refrigerate, covered, for at least 2 hours.
Preheat oven to 325 degrees. Coat a large piece of parchment with cooking spray. Cover strata with parchment, then with foil. Bake for 1 hour. Uncover and raise oven temperature to 375 degrees. Bake until top is golden brown and crusty, 20 minutes more. Let rest for 15 minutes. Serve warm.