Cut the baguette into ½-inch slices on the diagonal.
Using a baking dish, whisk together the eggs, half and half, sugar, 3 tablespoons of the rum, add in the slices of baguette, and soak for about 10 minutes turning occasionally until the liquid has been soaked up into the bread.
Heat a large skillet on medium heat and melt 1 tablespoon of butter. Place half of the bread into the skillet, cook until golden brown, and cooked through, about 4 minutes per side. Cook the other half of the bread using the other 1 tablespoon of butter.
In a small bowl, stir together the maple syrup and remaining rum, and place into the microwave to heat until warm, about 1 minute. Serve the French toast with the maple rum syrup and your favorite ice cream.