Preheat oven to 350°F and spray an 8-inch square metal baking pan with nonstick spray; dust with flour. Whisk together flour, oats, baking powder, cinnamon, allspice, cardamom, nutmeg, and salt in a medium bowl.
Beat butter, almond paste, and sugars in a large mixer bowl at medium speed until light and fluffy. Add eggs one at a time and beat well. Reduce speed to low and add half of flour mixture. Mix in milk and apples, then add remaining flour mixture.
Pour batter into prepared pan and smooth with an offset spatula; tap pan sharply to reduce air bubbles. Bake for 45 minutes, until cake is golden and a toothpick inserted in the center comes out clean or with a few crumbs. Let cool completely in pan on a wire rack.