To Make the Ice Cream
In a bowl, whisk together the egg yolks, ¼ cup of the sugar, and the salt. Combine the milk, cream, and remaining ¼ cup sugar in a heavy saucepan. Cook over medium heat, stirring occasionally, until almost simmering. Slowly pour the milk and cream into the egg mixture, whisking as you pour. Return the cream mixture to the saucepan. Cook over medium-low heat, stirring constantly with a heat-resistant plastic or wooden spatula, until the custard reaches 175°F and lightly coats the spatula. Whisk in the white chocolate until melted.
Strain the custard into a clean bowl and cool over an ice bath until room temperature. Refrigerate the custard for at least 4 hours or up to overnight. Place the dark and milk chocolates in a bowl and place the bowl in the freezer. Churn the ice cream base in an ice cream machine according to the manufacturer’s instructions. Fold the ice cream into the chocolate. Freeze until scoopable, about 4 hours, depending on your freezer.
To Make the Brownies
Preheat the oven to 350°F. Grease the bottom of a 9-by-13-inch baking pan and line it with parchment paper. Melt the chocolates and butter in a double boiler over hot water. In a medium bowl, whisk together the eggs and sugar. Whisk in the melted chocolate mixture. Sift together and then stir in the flour, salt, baking powder, and cocoa powder. Spread the batter in the prepared pan.
Bake until a skewer inserted in the center comes out fudgy and not dry, about 20 minutes. Let cool to room temperature. Run a knife around the inside edge of the pan. Place a cutting board on top of the pan. Invert the pan and board. Remove the pan and carefully peel off the parchment paper.
The brownies can be made up to 2 days in advance. Store at room temperature, wrapped in plastic wrap.