Preheat your oven to 400 F and line a large baking sheet with parchment paper. Combine the water, milk, butter, sugar, and salt in a medium saucepan over medium heat. Bring to a full rolling boil, making sure the butter is completely melted.
Remove from heat and add the gluten free flour all at once. Stir vigorously with a wooden spoon until the dough forms a smooth ball and pulls away from the sides of the pan. Return to medium-low heat and stir constantly for 1 to 2 minutes to dry the dough slightly.
Transfer the dough to a mixing bowl and let it cool for 5 minutes. Add the eggs one at a time, beating well with a hand mixer after each addition. After all three eggs, the dough should be smooth, glossy, and fall from a spoon in a thick V-shape.
Pipe or spoon mounds of dough onto the prepared baking sheet, about 1.5 inches wide and spaced 2 inches apart. You should get about 16 to 18 profiteroles. Lightly smooth any peaks with a wet fingertip.
Bake at 400 F for 15 minutes, then reduce the temperature to 350 F and bake for another 15 to 18 minutes until deeply golden, puffed, and hollow-sounding when tapped. Pierce a small hole in the bottom of each to release steam. Let cool completely on a wire rack.
Make the chocolate sauce by heating the heavy cream in a small saucepan until it just begins to simmer. Pour it over the chopped chocolate in a bowl, let sit for 1 minute, then stir until smooth. Stir in the butter and corn syrup until glossy.
Slice each profiterole in half horizontally. Place a scoop of vanilla ice cream inside each one and replace the top. Arrange on a serving plate and drizzle generously with the warm chocolate sauce. Serve immediately.