In a medium saucepan, combine diced apples, 1 cup white sugar, cornstarch, 1 teaspoon ground cinnamon, and 2 tablespoons water over medium heat.
Cook, stirring often, until the apples are tender and the mixture is thick and glossy, 5 to 7 minutes, then remove from heat and let cool to room temperature.
In a small bowl, stir together 1/4 cup white sugar and 1 teaspoon ground cinnamon; set aside for the cinnamon sugar swirl.
Preheat oven to 350°F. Grease a 9x13 inch baking pan with unsalted butter.
In a large mixing bowl, beat 1/2 cup unsalted butter and dark brown sugar with a hand mixer or stand mixer on medium speed until light and fluffy, 3 to 5 minutes, scraping the bowl as needed.
Add eggs one at a time, mixing well after each addition, then blend in 1 teaspoon vanilla bean paste and the Greek yogurt until smooth and creamy.
In a separate bowl, whisk together flour, baking powder, baking soda, and salt.
Add the dry ingredients to the wet ingredients in two additions, mixing on low speed just until no dry flour remains and a thick batter forms.
Spread half of the batter evenly in the prepared pan.
Spoon half of the cooled apple mixture over the batter in an even layer, then sprinkle with half of the cinnamon sugar mixture.
Gently spread the remaining batter over the apples, then top with the remaining apple mixture and the rest of the cinnamon sugar.
Bake for 35 to 40 minutes, until the top is golden, the center springs back lightly when touched, and a toothpick inserted in the center comes out clean or with a few moist crumbs.
While the cake bakes, prepare the glaze by whisking powdered sugar, milk, and the remaining 1 teaspoon vanilla bean paste in a medium bowl until smooth and pourable.
When the cake is done, place the pan on a wire rack and let cool for 10 to 15 minutes.
Using a toothpick or skewer, poke holes all over the warm cake, then slowly pour the glaze evenly over the surface, allowing it to soak in.
Let the cake cool until the glaze is set, then slice into squares and serve warm or at room temperature.