Preheat the oven to 350°F. Grease a 9x5 inch loaf pan and line it with a parchment sling for easy removal.
Toss the diced apples with the sugar and cinnamon in a bowl. In a separate small bowl, mix the brown sugar and cinnamon for the cinnamon layer.
In a medium bowl, whisk together the flour, baking powder, baking soda and salt.
In a large bowl, beat the softened butter and sugar with a mixer for 2-3 minutes, until light and fluffy. Beat in the sour cream and vanilla until smooth.
Add the dry ingredients in two additions, alternating with the milk, and mix on low just until combined. The batter will be thick, and the sour cream is what keeps this eggless loaf moist and tender.
Spread half of the batter in the pan, then top with half of the apples and half of the cinnamon layer mixture. Spoon the remaining batter over the top and gently spread it out.Scatter the remaining apples over the batter, pressing them in lightly, and sprinkle with the rest of the cinnamon layer mixture. Drag a skewer through the loaf in a few zigzags to create swirls.
Bake for 55-65 minutes, until a toothpick inserted in the center comes out free of raw batter. If the top browns too quickly, loosely tent it with foil after 35 minutes.Let the bread cool in the pan for 20 minutes, then lift it out onto a wire rack.
Whisk the powdered sugar, milk and vanilla into a smooth glaze. Pour it over the warm loaf so it drips into all the cracks, then let it set for 15 minutes before slicing.