Heat the oven to 350°F (175°C). Grease a 9-by-13-inch metal baking pan. Peel, core and dice the apples into roughly ¼-inch pieces.

Toss the apples with 1 teaspoon of the cinnamon. In a separate small bowl, mix the brown sugar, remaining cinnamon and melted tablespoon of butter to a damp, crumbly mixture.

Whisk the flour, baking powder, baking soda and salt in a bowl. In another bowl, beat the softened butter and granulated sugar until pale and fluffy, about 2 to 3 minutes. Beat in the eggs one at a time, then the vanilla.

Mix the sour cream and milk. On low speed, add the flour mixture to the butter mixture in three additions, alternating with the sour cream mixture. Stop as soon as combined. Fold in the cinnamon apples with a spatula.

Spread half the batter in the pan. Scatter half the brown-sugar mixture over it. Add the remaining batter in small dollops and gently spread, then scatter the remaining brown sugar on top. Run a knife through the batter in a few loose passes; do not blend the layers completely.

Bake for 40 to 50 minutes, beginning checks at 40 minutes. The top should be golden and a skewer in the center should come out with moist crumbs but no wet batter. Check a second spot if the first hits an apple piece.

Cool in the pan for about 20 minutes. Whisk the powdered sugar, milk and vanilla into a thick but pourable glaze. Drizzle over the warm cake, then wait at least 10 minutes for the glaze to settle. Cut into 16 squares.