Preheat the oven to 350 degrees F. Whisk 4 eggs, 1 1/2 cups half-and-half or milk, 1/2 teaspoon nutmeg, 1 teaspoon vanilla extract, 1 teaspoon sugar and a pinch of salt in a shallow bowl. Heat 1/2 tablespoon butter in a nonstick skillet over medium heat. Working in batches, dip 6 thick slices white bread, challah or brioche in the egg mixture and cook in the skillet until golden, 2 to 3 minutes per side, adding more butter as needed. Transfer to a baking sheet and bake until puffed, 8 to 10 minutes. Meanwhile, melt 4 tablespoons butter in the same skillet until browned, about 5 minutes. Drizzle the French toast with the browned butter and maple syrup.