Preheat the oven to 350°F (180°C). Butter and line an 8-inch (20-cm) square cake pan with parchment. Set aside.

To Make the Streusel Topping: In the bowl of a food processor, add the walnuts, oats, brown sugar, cinnamon, and cardamom and pulse until the mixture is evenly crumbly. Transfer to a bowl, then stir in the melted butter until well combined. Set aside.

To Make the Apple Cake: In the bowl of a stand mixer fitted with a paddle attachment or with a handheld electric mixer, beat the eggs and granulated sugar together on medium speed until light and fluffy, about 5 minutes.

Beat in the grated apple, butter, oil, vinegar, and vanilla.

In a separate bowl, combine the flour, baking soda, cinnamon, cardamom, and salt, then fold into the wet ingredients until just combined.

To Assemble and Bake: Spread the batter in the prepared pan, then sprinkle the streusel evenly over the top. Pat the streusel gently to ensure that it will stick to the cake.

Bake for 20-25 minutes, until a wooden skewer inserted in the center comes out clean.

Let the cake cool at least 20 minutes before lifting it out of the pan to cool completely on a wire rack.

Store leftovers in an airtight container at room temperature for up to 3 days.