For the cake: Separate the eggs. Place the egg whites in a medium bowl and yolks in a small bowl. Let stand at room temperature, about 20 minutes.
Preheat the oven to 350 degrees F and line the bottom of 3 (9-inch) round cake pans with parchment paper. Spray with nonstick baking spray. Set aside.
Sift together the sifted flour, baking powder, and 1/4 teaspoon salt in a large bowl. Add the brown sugar. If there are any large clumps of brown sugar, break them up with a spoon. Stir, then set aside.
Whip the egg whites to soft peaks with an electric mixer. With the mixer running, slowly add 1/4 cup sugar and continue to whip to stiff peaks. Add the egg yolks, half-and-half, oil, and vanilla to the flour mixture. Mix until just combined.
Fold the whipped egg whites into the cake batter a third at a time. Divide the batter among the prepared cake pans and bake for 18 to 20 minutes or until a toothpick inserted in the center of the cakes comes out clean. Let cool.
For the frosting: Place 4 tablespoons of water in a small bowl or ramekin. Sprinkle the gelatin powder over the top, making sure the gelatin is moistened. Let stand 5 minutes to bloom.
In the meantime, place the brown sugar, half-n-half, and 1/2 cup water in a medium saucepan. Stir. Cook over medium-low heat, stirring occasionally until the sugar is dissolved, about 5 minutes. Turn heat to low and add the bloomed gelatin. Stir. Cook 2 to 3 minutes, until the gelatin has dissolved. Remove from heat to cool to room temperature.
Whip the butter in a standing mixer until fluffy. Add the cooled gelatin mixture. Whip until thickened to a creamy frosting. This will look like a curdled hot mess but keep whipping and in 10 to 15 minutes it will become a beautiful buttercream. Mix in the vanilla.
For the butterscotch sauce:
Melt the butter in a 2-quart saucepan over low to medium heat. Add the brown sugar, lemon juice, and 1/4 teaspoon salt. Cook, stirring occasionally, until the sugar is melted and mixture is bubbling, about 5 minutes.
Remove the pan from the heat and add the heavy cream. The mixture will bubble. Whisk until incorporated. Return the pan to low heat and cook for 10 minutes without stirring, until slightly thickened. Remove pan from the heat and add the vanilla. Stir. Let cool to room temperature. Sauce will thicken as it cools.
Fill and frost the cooled cakes, drizzling the butterscotch sauce between the cake layers and on top of the frosting, and on top of cake.