Position racks in the upper and lower thirds of the oven, put a baking sheet on each rack, and heat the oven to 250°F.
Melt the butter in a small saucepan over medium heat. In a medium bowl, combine the melted butter, milk, eggs, sugar, vanilla, cinnamon, and 1 tsp. salt and whisk until the sugar and salt are dissolved. Transfer the mixture to a large baking dish.
Working in batches, add 2 or 3 slices of bread (or as many as will fit in your skillet in a single layer) to the mixture and soak, turning once, until saturated but not falling apart, about 2 minutes total.
In a 12-inch skillet over medium heat, melt about 1/2 Tbs. butter. When the foam subsides, use a slotted spatula to add the soaked bread in a single layer. Cook, turning once, until goldenbrown, 1-1/2 to 3 minutes per side. Transfer to a baking sheet in the oven, arranging the pieces in a single layer, to keep warm.
Repeat with the remaining bread, briefly rewhisking the batter before soaking, and wiping out the skillet and adding fresh butter between batches.
Serve drizzled with maple syrup.
The batter can be made ahead and kept at room temperature for up to 1 hour.