Place a piece of parchment in an 8-inch cake pan. Grease the parchment and sides of cake pan with butter.
In a large bowl, using an electric hand mixer on medium speed, cream the butter and sugar until light and fluffy, about 3 minutes. Add the eggs one at a time, beating until combined after each and scraping down the sides and bottom of the bowl as needed, Beat in the vanilla extract.
In a medium bowl, whisk together the oat and sweet white rice flours with the baking powder and salt.
With the mixer on low, stir half of the flour mixture into the butter mixture until just combined. Stir in the milk until just combined, then add the rest of the flour, scraping down the sides and bottom of the bowl as needed.
Position a rack in the center of the oven and heat to 350º F.
Spread the batter evenly in the pan and let rest for at least 15 minutes. Then preheat oven to 350°F.
Place the cake in the oven and bake until the top is golden and a toothpick inserted near the center comes out clean, the top is set, and the edges are slightly coming away from the pan, 40-50 minutes. Remove from oven and let cake cool completely in the pan on a rack, then invert onto a serving platter and peel away the parchment.
Using a serrated knife, split cake in half horizontally; place bottom half, cut side up, on a serving plate.