Preheat oven to 350°F (175°C) with a rack in the center. Line a 9x5-inch loaf pan with parchment paper, leaving an overhang on the long sides, and lightly grease any exposed pan surface.
In a small bowl, stir together ⅓ cup (67 g) granulated sugar and 1 tablespoon ground cinnamon until evenly combined. Set aside.
Peel, core, and dice the apples into ¼-inch (6 mm) pieces. Toss them with 1 tablespoon (8 g) flour until every piece is lightly coated so they stay suspended in the batter.
In a large bowl, whisk together the 2 cups (250 g) flour, baking powder, baking soda, salt, 2 teaspoons cinnamon, and nutmeg until the leavening is evenly distributed.
In a separate bowl, whisk the melted butter, granulated sugar, and brown sugar until smooth. Add the eggs one at a time, whisking after each, then whisk in the vanilla and buttermilk until uniform.
Pour the wet ingredients into the dry ingredients and fold gently with a silicone spatula just until no dry streaks remain. Fold in the flour-coated apples with two or three strokes; a few small lumps are fine.
Spoon half the batter into the prepared pan and smooth it level. Sprinkle about two-thirds of the cinnamon sugar over the batter, spoon the remaining batter on top, smooth it, then sprinkle the last of the cinnamon sugar over the surface. Drag a butter knife through in a few S-shaped passes to swirl.
Bake for 55 to 65 minutes, until the top is golden and a toothpick inserted into the center comes out with only a few moist crumbs. If the top browns too fast around 40 minutes, tent loosely with foil for the rest of the bake.
Cool the loaf in the pan for 15 minutes, then lift it out using the parchment overhang onto a wire rack. Cool at least another 45 minutes before slicing so the crumb can set.
Once the loaf is fully cool, whisk the confectioners' sugar, milk, and ½ teaspoon vanilla into a smooth, pourable glaze. Drizzle over the top and let it set for a few minutes before serving.