Preheat oven to 350°F (175°C) and line two 9-inch round cake pans with parchment paper, then grease the sides.

Whisk together 2 1/4 cups flour, 2 1/2 teaspoons baking powder, and 1/2 teaspoon salt in a medium bowl.

In a large bowl, beat 1/2 cup softened butter and 1 1/2 cups sugar at medium speed for 3 minutes until light and fluffy.

Add eggs one at a time, beating after each addition, then mix in 1 1/2 teaspoons vanilla extract until smooth.

Alternately add the flour mixture and 1 cup milk to the batter in three additions, beginning and ending with flour, mixing until just combined.

Divide batter evenly between pans and smooth tops; bake for 28–32 minutes until a toothpick comes out clean and tops spring back.

While cakes bake, combine 14 ounces sweetened condensed milk, 6 tablespoons butter, 3/4 cup brown sugar, and 1 teaspoon vanilla in a saucepan over medium heat and cook for 5–7 minutes, stirring constantly, until glossy and slightly thickened.

Let cake layers cool in pans for 10 minutes, then transfer to wire racks and cool completely, about 30 minutes more.

Place one layer on a serving plate, pour half of the warm filling over it, stack the second layer, then pour remaining filling over top and sides; let rest 1 hour before slicing.