Using caramel bars (heavy stainless-steel bars), make a square frame measuring 30cm/12in long on each side by 2.5cm/1in deep over a flat sheet pan lined with a nonstick rubber mat . Freeze this setup.
Warm up the first amount of heavy cream in a sauce pot.
In 3.76-l/4-qt sauce pot, cook the sugar over high heat, stirring constantly until it turns a medium amber brown, 170°C/338°F. Stir in the butter completely, then add the warm heavy cream. Remove the caramel from the heat.
Place the eggs over a hot water bath and warm them to 60°C/140°F while stirring constantly. Transfer them to the bowl of an electric mixer and whip until they have cooled to room temperature. While the eggs are whipping, stream in the caramel. Add the gelatin and whip until the gelatin dissolves and the mixture has cooled to 21°C/70°F.
Whip the second amount of heavy cream and the stabilizer in an electric mixer to medium-soft peaks. Fold the whipped cream into the caramel in 2 additions.
Pour into the prepared frame. Even out the top of the frame. Freeze to harden.
Once hardened completely, remove the frame and then flip the mousse over onto a cutting board lined with parchment paper. Cut the mousse into 4 even squares measuring 15 cm/6 in long on each size using a warm slicing knife. Dip the knife in hot water and then dry it to get a clean cut.