Instructions

for the apple filling

Adjust oven rack to middle position and heat oven to 350 degrees. Make parchment sling for 9-inch square baking pan by laying 2 long sheets of parchment so each is 9 inches wide. Lay sheets in pan perpendicular to each other, with extra parchment hanging over the edges of the pan. Grease parchment.
Melt 4 tablespoons unsalted butter in 12-inch skillet over medium-high heat. Stir in 4 large Honeycrisp apples (peeled, cored and sliced very thin), ½ cup (3½ ounces) granulated sugar, 2 teaspoons ground cinnamon, and ½ teaspoon table salt and cook, stirring occasionally, until apples are just tender, about 7 minutes. Whisk 3 tablespoons lemon juice and 1 tablespoon superfine brown rice flour together in small bowl. Stir lemon juice mixture into apples and cook until thickened, about 30 seconds. Transfer to large bowl; set aside.

for the cake

Stir together 1¼ cups (5 ounces) almond flour, 1 cup (4½ ounces) superfine brown rice flour, 2 teaspoons baking powder, 1 teaspoon ground cinnamon, 1 teaspoon ground ginger, and ½ teaspoon table salt in large bowl. Whisk 1 cup (7 ounces) packed light brown sugar, ⅔ cup vegetable oil, ½ cup Greek yogurt, 2 large eggs, and 2 teaspoons vanilla extract together in second large bowl until smooth. Stir sugar mixture into flour mixture until smooth.

for the crumble

Stir ⅔ cup finely chopped walnuts, ¼ cup (1¾ ounces) packed light brown sugar, 3 tablespoons superfine brown rice flour, and ⅛ teaspoon table salt together in medium bowl. Add ¼ cup chilled unsalted butter cut into ½-inch pieces, and, using your fingers, work butter into walnut mixture until crumbly and sandy dough forms.
Spread half of cake batter evenly in prepared pan. Top with half apple filling followed by remaining half of cake batter and remaining apple filling. Sprinkle crumble over top.
Bake until crumble is golden brown and toothpick inserted in center comes out clean, 45 to 50 minutes. Let cool in pan for at least 2 hours. Carefully remove from pan and cut into squares. Serve.