Chocolate Ice Cream Sandwiches

2½ ounces semisweet or bittersweet chocolate, roughly chopped (about ½ cup plus 1 tablespoon)
1½ ounces unsweetened chocolate, roughly chopped (about 1/3 cup)
3 tablespoons unsalted butter, cut into small pieces, plus more for greasing the pan
1 large egg
1/3 cup sugar
1 teaspoon vanilla extract
1 tablespoon plus 2 teaspoons all-purpose flour
1/8 teaspoon baking powder
1/8 teaspoon fine sea salt, plus an extra dash
1 pint ice cream  

1. Preheat the oven to 350°F. Grease two baking sheets with butter.
2. In a medium bowl, combine the chocolates and butter. Create a double boiler by filling a medium pot that will hold the bowl of chocolate snuggly with 1 inch of water. (The bowl should not touch the water.) Bring the water to a boil, then lower the heat so the water is hot but not boiling. Set the bowl of chocolates and butter on the pot, stirring often with a rubber spatula, until melted. Remove from the heat. 
3. In the bowl of an electric mixer fitted with a whisk attachment, beat the egg on high until fluffy, about 1 minute. Add the sugar and vanilla and beat on high until the mixture thickens slightly and turns pale yellow, about 5 minutes. (Stop to scrape down the sides of the bowl.) Using a rubber spatula, gently mix in the chocolate mixture. Sift the flour, baking powder, and salt together. Slowly add the flour mixture to the chocolate mixture and stir until combined.
4. Drop 1 tablespoon lumps of the cookie batter onto the baking sheets, leaving about 1½ inches between each. You should have about 10 cookies per baking sheet. Bake for 4 minutes, rotate the pan from front to back, and bake until they crack slightly but still look somewhat moist, an additional 4 minutes. Let the cookies cool for 10 minutes on the sheets, then transfer to a rack to cool completely. (They will flatten slightly once cooled.)
5. Scoop 3 tablespoons of ice cream and press into a fairly round 1-inch-thick puck. Sandwich the ice cream puck between 2 cookies and press gently until the ice cream is about ¾ inch thick. Eat immediately or wrap and freeze until ready to eat. Repeat with the remaining cookies and ice cream. Makes about 10 mini (2-inch) sandwiches. Adapted from “Old School Comfort Food” by Alex Guarnaschelli.