Preheat oven to 350°F (175°C). Lightly grease a 9×13-inch baking dish.
In a bowl, cream butter and sugar until light. Add eggs one at a time, mixing well after each.
Mix in vanilla extract, then gradually add sifted flour, baking soda, and salt until smooth.
Spread the batter evenly in the prepared pan and bake for 20-22 minutes until the center is just set.
In a small saucepan, cook heavy cream, brown sugar, 1/2 teaspoon salt, and 1-2 tablespoons of the cake batter (to add gooey texture) over medium heat. Boil for 1 minute, then cool slightly.
Drizzle the caramel over the warm dessert, sprinkle remaining sea salt flakes, and let sit 10 minutes before serving.