Prepare the apple filling. Melt the butter in a skillet over medium-low heat. Add the shredded apples, maple syrup, cinnamon, salt, lemon juice, and brown rice flour. Stir and cook for 2–3 minutes, until slightly softened and thickened. Set aside.
Prepare the cake batter. Preheat the oven to 350°F. In a large bowl, whisk together the brown rice flour, arrowroot flour, almond flour, baking powder, cinnamon, ginger, and salt. In a separate bowl, whisk the sugar, oil, yogurt or buttermilk, eggs, and vanilla. Add the wet ingredients to the dry ingredients and mix until just combined.
Prepare the crumble. Combine the walnuts, sugar, brown rice flour, and salt in a small bowl. Cut in the chilled butter using your fingers or a pastry cutter until the mixture forms coarse crumbs.
Layer the cake. Grease an 8x8-inch or 9x9-inch baking pan. Spread half the cake batter in the pan, followed by the apple filling. Gently spread the remaining batter over the apples, then sprinkle the crumble evenly on top.
Bake. Bake at 350°F for about 30 minutes, or until the top is golden and a toothpick inserted into the cake portion comes out clean or with a few moist crumbs. If needed, cover loosely with foil to prevent overbrowning.
Cool and serve. Let the cake cool in the pan for 15–20 minutes before slicing. Serve warm or at room temperature.