In a light-colored saucepan, melt ½ cup (113g) unsalted butter over medium heat. Stir often as it foams, sizzles, and then quiets. Cook until golden brown specks form on the bottom and the butter smells nutty. Remove from heat and pour into a heatproof bowl to cool for 10 minutes.

In the same skillet used to brown butter, melt 2 Tbsp butter over medium heat. Add diced apples, brown sugar, cinnamon, and lemon juice (or vinegar). Cook 3–4 minutes, until just softened. Remove from heat, stir in 1 Tbsp flour, and set aside to cool slightly.

In a small bowl, combine brown sugar, cinnamon, salt, and cardamom (if using).

Preheat oven to 350°F (177°C). In a large bowl, whisk cooled browned butter with granulated sugar until combined. Add eggs one at a time, whisking well. Stir in vanilla, milk, and sour cream until smooth. In another bowl, whisk together flour, baking powder, baking soda, and salt. Gently fold dry ingredients into wet ingredients until just combined. Do not overmix.

Spread half the batter into a greased and lined 9×5-inch loaf pan. Spoon half the apple mixture over the batter, try to keep the apples in the center of the batter and away from the edges, then sprinkle half the cinnamon sugar mixture. Repeat layers with remaining batter, then apples, and top the loaf off with cinnamon sugar.

Bake at 350°F (177°C) for 50–60 minutes, or until a toothpick inserted in the center comes out mostly clean (a few moist crumbs are okay). If the top browns too quickly, cover loosely with foil for the last 15 minutes.

Let loaf cool in pan for 15 minutes, then transfer to a wire rack. Whisk powdered sugar, milk, vanilla, and salt until smooth. Drizzle over cooled loaf. Slice and enjoy