With the rack in the middle position, preheat the oven to 180 °C (350 °F). Butter two 20-cm (8-inch) springform pans. Line the bottom with parchment paper.
In a bowl, beat the egg whites with an electric mixer until soft peaks form. Gradually add 125 ml (1/2 cup) of sugar and beat until stiff peaks form. Set aside.
In another bowl, combine the oil, egg yolks, and vanilla. Set aside.
In a large bowl, combine the flour, remaining sugar (375 ml / 1 1/2 cups), baking powder, baking soda, and salt. Make a well. Add the oil and egg mixture and hot milk, stirring with a whisk until smooth. With a spatula, gently fold in the meringue. Spoon the batter into the prepared pans. Bake for about 45 minutes or until a toothpick inserted in the centre of the cake comes out clean. Let cool. Unmould and cool completely on a wire rack.
Place the eggs in large bowl. Set aside.
In a saucepan, bring the water, sugar, and syrup to a boil. Cook until golden brown. Add the cream and stir to combine.
Beat the eggs with an electric mixer at medium speed until light and fluffy. Drizzle the hot syrup over the eggs, avoiding the beaters of the mixer. Beat continuously until completely cooled, about 15 minutes. Add the butter, one or two pieces at a time, beating until the buttercream is very smooth. Set aside.
Line a 43 x 30-cm (17 X 12-inch) baking sheet with parchment paper.
In a saucepan, bring the water, sugar, butter, and salt to a boil. Cook until the mixture caramelizes and take on a golden brown colour. Remove from the heat immediately and stir in the baking soda with a whisk. Quickly pour onto the baking sheet. With a spatula, spread the caramel in a thin layer. Cool completely and break into long pieces.
In a bowl, beat the dulce de leche to soften, if necessary. Set aside.
Cut the cake tops to make them flat, if necessary. Cover one cake with a third of the buttercream. Drizzle with half of the dulce de leche. Top with the second cake. Frost the entire cake with the buttercream and drizzle the remaining dulce de leche on top of the cake. Press the pieces of caramel brittle all around the cake, overlapping them slightly.
Dulce de leche is sold in the caramel spread and jams aisle in most grocery stores.
Do not store the cake in the refrigerator. Leave it at room temperature. In warm weather, store in a cold room or in the coolest room of the house.
Recipes with the "Healthy Pick" stamp have been evaluated by a registered dietitian member of the Ordre professionnel des diététistes du Québec.