Heat the oven to 375°F. Line a large rimmed baking sheet with aluminum foil.

Whisk 6 tablespoons melted unsalted butter, ½ cup of the granulated sugar, and ½ cup maple syrup together in a large bowl until a paste forms. Scrape onto the baking sheet and spread into an even layer.

Whisk 5 large eggs, 1 ¾ cups dairy of choice, the remaining ¼ cup granulated sugar, 2 tablespoons orange liqueur if using, 2 teaspoons vanilla extract, ½ teaspoon ground cinnamon, and ¼ teaspoon kosher salt together in the now-empty bowl until fully combined, no streaks of egg remain, and the sugar is dissolved.

Cut 1 loaf bread crosswise into ¾-inch to 1-inch-thick slices. Working 1 slice at a time, soak the bread slices until drenched but not falling apart, 10 to 15 seconds per side. Place in a single layer on the baking sheet.

Bake until the tops are golden-brown and the sugar paste is bubbling, 25 to 30 minutes. Remove from the oven. Turn the oven to broil. Use a flat spatula to flip the slices over. Return to the oven and broil, rotating the pan halfway through for even browning, until bubbling and darkened slightly in color, 2 to 4 minutes total. Let cool slightly to allow the top to crisp up before serving.