2 tablespoons unsalted butter, plus more for the baking dish
3 cups whole milk, or more if needed
2 tablespoons all-purpose flour
6 ounces Monterey Jack cheese, grated (1 1/2 cups)
6 ounces fresh goat cheese, cut into small pieces
1/2 cup freshly grated Parmesan cheese
Kosher salt and freshly ground black pepper
1 medium head of cauliflower, cut into florets, each floret cut into 2 or 3 pieces
Chopped fresh flat-leaf parsley, for garnish