Preheat the oven to 250 degrees F. Cut the baguette on the diagonal into eight 1/2-inch-thick slices. Whisk the eggs, half-and-half, sugar and 3 tablespoons rum in a large baking dish. Add the bread and soak, turning occasionally, about 10 minutes. (Remove before the bread is soggy.)
Melt 1 tablespoon butter in a large nonstick skillet over medium heat. Add half of the bread and cook until golden brown and cooked through, about 4 minutes per side. Transfer to a baking sheet and keep warm in the oven. Add the remaining 1 tablespoon butter to the skillet and repeat with the remaining bread.
Meanwhile, mix the maple syrup and the remaining 1 tablespoon rum in a small microwave-safe bowl; microwave until warm, about 1 minute. Serve the French toast with the maple-rum syrup and ice cream.