Heat the oven to 350°F with a rack in the middle position. Set a wire rack in a rimmed baking sheet. In a medium bowl, stir together the apples, lemon juice, 1 tablespoon of the white sugar and ½ teaspoon of the cinnamon. Set aside.

In a second medium bowl, whisk together both flours, the baking powder, salt, allspice, remaining 1½ teaspoons cinnamon and the remaining 4 tablespoons white sugar.

In a third medium bowl, whisk together the eggs, milk, ½ cup of the sour cream and the vanilla. Add the milk mixture to the flour mixture and whisk just until the batter is evenly moistened; do not overmix.

Drain the apples in a colander and transfer to a paper towel–lined plate; discard the liquid. With additional paper towels, press the apples as dry as possible; return them to the bowl.

Heat a 12-inch nonstick skillet over medium-low until hot, 3 to 4 minutes. Add ½ teaspoon oil to the skillet, then use a wad of paper towels to spread an even film of it over the skillet, wiping out any excess.

Using 2 heaping tablespoons of batter per pancake, form 3 pancakes in the pan, using the back of a spoon to spread each into a 4-inch round. Working quickly, scatter about 1 heaping tablespoon of apple cubes onto each and gently press into the batter.

Cook until the edges begin to look set and the bottoms are golden brown, 2 to 3 minutes; lower the heat if they brown too quickly. Using a wide spatula, flip each pancake and cook until the second sides are well browned, another 1 to 2 minutes.

Transfer the pancakes apple-side down to the wire rack. Repeat with the remaining batter, coating the skillet with oil for each batch.

Place the baking sheet with the pancakes in the oven and heat until warmed through, about 5 minutes. Transfer to plates. Put a spoonful of powdered sugar in a fine mesh strainer and lightly dust the pancakes. Serve warm, passing the remaining ½ cup sour cream at the table.