Add the butter to a light-colored saucepan or skillet over medium heat.

Cook, stirring occasionally, as the butter melts and begins to foam.

Continue cooking until the milk solids at the bottom of the pan turn golden brown and the butter smells nutty and toasted.

Immediately pour the browned butter into a heatproof bowl, scraping all of the browned milk solids from the pan into the bowl.

Add the diced apple, flour, cinnamon, and granulated sugar to a small bowl.

Toss until the apple pieces are evenly coated. Set aside.

Preheat the oven to 350°F.

Grease a 9x5-inch loaf pan and line it with parchment paper, leaving enough overhang along the sides to easily lift out the baked loaf.

In a large mixing bowl or the bowl of a stand mixer, combine the cooled browned butter, granulated sugar, and dark brown sugar.

Beat on medium speed for about 2–3 minutes, until well combined.

Add the eggs one at a time, mixing well after each addition. Add the egg yolk and mix until incorporated.

Add the sour cream, whole milk, and vanilla bean paste. Mix until combined.

In a separate bowl, whisk together the flour, cornstarch, cinnamon, nutmeg, baking powder, baking soda, and kosher salt.

Add the dry ingredients to the wet ingredients and mix on low speed just until combined. Do not overmix.

Gently fold the cinnamon sugar apples into the batter until evenly distributed.

Transfer the batter to the prepared loaf pan and smooth the top.

Bake for 60–70 minutes, beginning to check the loaf around the 50-minute mark.

The loaf is done when a toothpick inserted into the center comes out without wet batter. A few moist crumbs are fine. If you hit an apple piece, test another area.

If the top becomes darker than you'd like before the center finishes baking, loosely tent the loaf with aluminum foil.

Let the loaf cool in the pan for about 15 minutes, then use the parchment paper to carefully transfer it to a wire rack.

While the loaf bakes, combine the apple cider, granulated sugar, and kosher salt in a small saucepan.

Bring to a gentle simmer over medium heat.

Simmer for about 5–8 minutes, until the cider has reduced slightly and the apple flavor becomes more concentrated.

Remove from the heat and let cool slightly.

While the loaf is still warm, gradually brush the cider syrup over the top, allowing it to absorb between additions. You may not need to use all of the syrup.

Allow the loaf to cool completely before icing.

Add the sifted powdered sugar, cinnamon, vanilla bean paste, and salt to a medium bowl.

Add about 2-3 tablespoons heavy cream and whisk until smooth.

Continue adding heavy cream about 1 tablespoon at a time, as needed, until the icing is thick but spreadable.

Spread the icing over the completely cooled loaf.

Allow the icing to set before slicing and serving.