Place 4 cups of water in a medium pot and bring it to a boil. When the water is boiling, slowly add polenta and salt while whisking. Turn the heat down and continue to stir every few minutes to make sure the polenta doesn’t stick to the bottom of the pot. As soon as the polenta gets thick, add the sugar. Stir every few minutes until the mixture thickens again. Then, add the cream. Stir until the liquid evaporates and the mixture thickens. Remove from the heat and stir to release the steam. Add the egg and stir quickly so that the egg doesn’t have time to cook (you don't want scrambled eggs in your polenta!). When the egg is incorporated, return the pot to low heat and add butter. As soon as the butter is melted, turn off the heat.
Prepare an ice bath by placing ice and cold water in a large bowl. Place another smaller bowl inside the ice water. Pour polenta into the smaller bowl, being careful not to allow water in the polenta. Cover with plastic wrap and allow to cool.
Wash and peel your apple, cut it in half and remove the core. Cut the apples into ½-inch cubes. In a small saucepan, melt butter. Add brown sugar and salt, then stir to combine. Add apple cubes and cook until soft.
In a separate bowl, combine the mascarpone, heavy cream, sugar, and vanilla extract and fold until well combined.
To plate, place polenta in a small bowl. Layer sautéed apple and mascarpone cream on top, then serve at room temperature.