Preheat the oven to 350°F and thoroughly grease a 9x13-inch baking dish with butter or nonstick spray.
In a small bowl, stir together the 1/3 cup granulated sugar and 1 tablespoon cinnamon for the swirl; set aside.
In a large bowl, beat the softened butter, granulated sugar, and brown sugar with a hand mixer on medium speed for 2 to 3 minutes until light and fluffy.
Add the eggs one at a time, beating well after each, then mix in the vanilla extract until combined.
In a separate bowl, whisk together the flour, baking soda, salt, and 2 teaspoons of cinnamon for the batter.
Add the dry ingredients to the butter mixture in three additions, alternating with the buttermilk, starting and ending with the dry ingredients. Mix on low speed just until combined, being careful not to overmix.
Using a rubber spatula, gently fold the diced apples and chopped walnuts into the batter until evenly distributed.
Spread half of the batter into the prepared pan, sprinkle with half of the cinnamon sugar swirl, then top with the remaining batter. Sprinkle the rest of the cinnamon sugar on top and drag a knife through the batter to swirl.
Bake for 35 to 40 minutes, until a toothpick inserted in the center comes out clean or with a few moist crumbs. Tent with foil if the top browns too quickly.
Cool the cake on a wire rack for at least 20 minutes. Whisk together the powdered sugar, milk, and vanilla for the glaze, then drizzle over the slightly warm cake before slicing and serving.