Preheat your oven to 350°F (175°C). Grease a 9x13 inch baking pan or line it with parchment paper, allowing some overhang for easy removal.
In a medium bowl, whisk together the flour, baking powder, and salt.
In a large bowl, cream together the softened butter and granulated sugar until light and fluffy, about 3-4 minutes.
Beat in the eggs one at a time, then add the vanilla extract. Mix until well combined, scraping down the sides of the bowl as needed.
Add the flour mixture to the wet ingredients alternately with milk, beginning and ending with the flour mixture (flour-milk-flour-milk-flour). Mix just until combined after each addition.
Gently fold in the diced apples with a spatula.
In a small bowl, combine the melted butter, brown sugar, flour, cinnamon, and nutmeg for the swirl. Stir until thoroughly mixed.
Pour half of the batter into the prepared pan, spreading it evenly with a spatula. Drizzle half of the cinnamon swirl mixture over the batter and use a knife to gently swirl it into the batter.
Add the remaining batter, then repeat with the remaining swirl mixture.
Bake for 30-35 minutes, or until a toothpick inserted into the center comes out clean with a few moist crumbs. The cake will be golden brown and slightly pull away from the edges when done.
Allow the cake to cool for about 15 minutes.
While the cake cools, prepare the glaze by whisking together powdered sugar, milk, vanilla, and salt until smooth. Adjust consistency if needed.
Pour the glaze over the warm (not hot) cake for that classic cinnamon roll finish.