Preheat the oven to 400 degrees F.
With a rubber spatula, scoop the pate a choux into the pastry bag and pipe out approximately 16 (1 1/2-inch rounds) spaced 1 to 2 inches apart on the parchment paper-lined baking sheet. Brush lightly with the beaten egg and place in the oven. Cook until golden and puffed, about 25 minutes. Remove from the oven and cool on a baking rack.
Split the profiteroles nearly in half, leaving a little hinge at the back, and cradle a small scoop of vanilla ice cream inside each. Divide the profiteroles between bowls and pour over some chocolate sauce. Serve cold.
In a small saucepan over high heat, bring the water, salt, sugar, and butter to a boil, making sure the butter is completely melted. Off the heat, add the flour all at once and beat vigorously with a wooden spoon. Return to the heat and continue beating until the dough forms a solid, smooth mass and pulls away from the sides of the saucepan. Take off the heat and empty the dough into a clean mixing bowl. Little by little add the beaten eggs, beating vigorously in between each addition, until the dough forms a smooth, supple mass. Divide the dough into 2 even quantities, 1 part to be used for the profiteroles, the other for gougeres.