Preheat the oven to 325°F (163°C). Line a 13x9-inch pan with parchment paper so it covers the bottom and two sides, then grease the pan with 2 tablespoons softened butter; set aside.
In a stand mixer fitted with the whisk (or using a hand mixer), beat 4 large room-temperature eggs on medium speed until fluffy and light yellow. Add 2 cups granulated sugar and beat until combined.
Add the melted 8 ounces (2 sticks) unsalted butter, 1 1/4 cups sifted unsweetened cocoa powder, 1 teaspoon vanilla extract, 1/3 cup sifted all-purpose flour, and 1/2 teaspoon kosher salt to the egg-sugar mixture and mix until just combined.
Pour the batter into the prepared pan and smooth the top. Bake for about 30–35 minutes, or until a toothpick inserted in the center comes out with a few moist crumbs attached. (Oven times vary.)
Remove the pan from the oven and place it on a cooling rack. Do not remove the brownies from the pan; let them cool completely in the pan.
While brownies cool, prepare the chocolate mousse: bring a small pot of water to a simmer. Place the 10 ounces finely chopped milk chocolate in a heatproof bowl and melt it over the simmering water, stirring until smooth, then remove from heat.
In a small bowl, combine 7 grams gelatin with 2 tablespoons water and let the gelatin bloom for about 10 minutes.
Heat 1/2 cup of the chilled whipping cream in a small saucepan until it just comes to a boil. Remove from heat and stir in the bloomed gelatin until dissolved and fully incorporated.
Pour about half of the warm cream-gelatin mixture into the melted chocolate and whisk quickly to combine, then add the remaining cream mixture and whisk until smooth and shiny. Set aside to cool to room temperature.
Place the remaining 2 cups chilled whipping cream in a clean bowl and whisk with the mixer until soft peaks form.
When the chocolate mixture is cooled to room temperature, gently fold the whipped cream into the chocolate mixture with a spatula until no streaks remain, creating a light mousse.
Pour the mousse evenly over the completely cooled brownies, smooth the top, then refrigerate the pan overnight or for at least 6 hours until the mousse is set.
To serve, lift the brownies from the pan using the parchment edges, dust with cocoa powder if desired, rinse a knife in cold water and dry it, then cut into 12 squares.