Preheat the oven to 350°F. Grease a 10- or 12-inch round baking dish.
In a large bowl, whisk together the milk, eggs, mustard, dried thyme, onion powder, and cayenne, and season with salt and pepper. Working in batches, submerge the croissants into the egg mixture, allowing them to soak for about 1 minute per side. Arrange the croissants in the prepared baking dish.
Add the broccoli and Gruyère to the remaining egg mixture in the bowl. Stir to combine, then pour it over the croissants. Arrange the ham over the top, then evenly scatter with the fontina.
Cover the baking dish with aluminum foil and bake until the cheese is melting and the egg mixture begins to set, about 30 minutes. Remove the foil, scatter the butter slices over the top, and continue baking, uncovered, until the croissants are golden and the cheese is completely melted, 15 to 20 minutes more.
Sprinkle with fresh thyme and serve warm, family style.