Irish Apple Cake

4 tablespoons butter
1 cup granulated sugar
1 egg, beaten
4 Granny Smith apples, cored, peeled, and diced
1/4 cup chopped walnuts
1 teaspoon vanilla extract
1/2 teaspoon baking powder
1/2 teaspoon baking soda
1/2 teaspoon salt
1/2 teaspoon ground cinnamon
1/2 teaspoon ground nutmeg
1 cup all-purpose flour
Whipped cream or vanilla ice cream for serving

Preheat the oven to 350°F. Generously grease an 8-inch square cake pan. In a large bowl, cream the butter and sugar together until light and fluffy. Add the egg, apples, nuts, and vanilla and stir well. Sift in the dry ingredients and mix well. Pour the batter into the prepared pan and bake until the cake is lightly browned and a skewer inserted into the center comes out clean, about 45 minutes. Let cool in the pan for 5 minutes, then unmold and serve warm or at room temperature with whipped cream or vanilla ice cream.Serves 10 to 12.