How to Make It
In a large saucepan, bring 1 3/4 cups of water to a boil with the butter and 1/2 teaspoon of salt. Remove from the heat, add the flour and stir vigorously until incorporated. Return to moderate heat and cook, stirring, until the dough pulls away from the side, 3 minutes. Remove from the heat.
Using an electric mixer, beat the dough at low speed for 1 minute, just until slightly cooled. Increase the speed to medium-high and beat in the eggs and egg white, one at a time. Transfer the dough to a bowl, press a sheet of plastic wrap directly on the surface and refrigerate until cooled to room temperature, about 15 minutes.
Meanwhile, in a large saucepan, heat the milk with 1/2 cup of water until small bubbles appear around the edge. Add the chocolate, cocoa powder, 1/2 cup plus 2 tablespoons of the sugar and a pinch of salt and whisk over low heat until the chocolate is melted. Keep warm over very low heat.
In another large saucepan, heat the oil to 375°. Set a rack on a large baking sheet and cover with paper towels. In a pie plate, combine the remaining 1/2 cup of sugar with the cinnamon. Scoop the dough into a large pastry bag fitted with a large star tip.
Working quickly, squeeze 6-inch lengths of the dough into the hot oil, cutting them off with a knife. Fry no more than 8 churros at a time; they expand a bit as they cook. Fry over moderately high heat, turning once or twice, until browned, about 8 minutes. Drain the churros on the rack for 2 minutes, then toss with the cinnamon sugar in the pie plate. Serve at once with hot chocolate.