1/4cupnonfat plain Greek yogurtat room temperature
1teaspoonpure vanilla extract
2cupspeeled and 1/4-inch-chopped applesabout 2 medium apples—choose a firm, flavorful variety such as Granny Smith, Fuji, McIntosh, or Cortland
For the Brown Sugar Cream Cheese Frosting:
6ounces reduced-fat cream cheeseat room temperature
2tablespoonsunsalted butter at room temperature
1/2 cupbrown sugar
Place the rack in the center of your oven. Preheat the oven to 325 degrees F. Line a 9x9-inch or 8x8-inch square baking pan with parchment paper so that it overhangs two of the sides like handles. Lightly coat the parchment with nonstick spray. Set aside. Place the nuts in a single layer on an ungreased baking pan. Bake until toasted and fragrant, about 8 to 12 minutes. Do not walk away during the last few minutes (this is when nuts love to burn). Immediately transfer to a cutting board. Coarsely chop and set aside.
In a medium bowl, whisk together the white whole wheat flour, baking soda, baking powder, salt, cinnamon, nutmeg, and allspice.
In the bowl of a standing mixer fitted with the paddle attachment or a large mixing bowl, cream together the butter, brown sugar, and granulated sugar until light and fluffy, about 3 minutes. Beat in the eggs, one at a time, stopping in between each to scrape down the bowl. Beat in the Greek yogurt and vanilla extract. With the mixer on low, slowly add the dry ingredients, mixing just until evenly moistened. Fold in the apples and 1/2 cup of the toasted chopped walnuts. The batter will be very thick.
Scrape the batter into the prepared pan and with the back of a spatula, spread it evenly. Bake until a toothpick inserted into the center comes out clean and the edges are light golden brown, about 35 to 45 minutes. The baking time will vary depending upon the moisture content of your apples. The center of the cake will still feel soft but should spring back just a little when touched. Check at the 35-minute mark. If your cake isn't done but the top is browning more quickly than you would like, loosely tent it with foil, then continue baking as directed. When it's done, place the pan on a wire rack and let cool completely before frosting.
To make the frosting: In a mixing bowl or a stand mixer fitted with the paddle attachment, beat the cream cheese, butter, and brown sugar at medium speed until creamy, about 1 to 2 minutes. Add the vanilla and salt and beat until blended. Reduce the mixer speed to low. Gradually add the powdered sugar, beating just until combined. Increase the mixer speed to high. Beat 1 additional minute or until smooth. Spread the frosting generously onto the cooled cake. Sprinkle the remaining walnuts over the top. Slice generously and serve.
TO STORE: Store leftovers in an airtight container in the refrigerator for up to 5 days. (I find it tastes even better and more moist on day two! Let come close to room temperature prior to serving.)
TO FREEZE: Freeze (frosted or unfrosted) for up to 1 month. Let thaw overnight in the refrigerator.