Preheat the oven to 350°F (180°C). Grease a 9-inch round baking dish with butter.
Separate the egg whites from the yolks. Place the whites in a clean, dry mixing bowl.
In a separate bowl, whisk together the oat flour, rice flour, and baking powder.
Combine the egg yolks and sugar in a large bowl. Beat with an electric mixer for about 5 minutes, until pale, thick, and fluffy.
Add the softened butter and beat again until well combined.
Gradually add the flour mixture. Mix on low speed just until incorporated.
Add a pinch of salt to the egg whites and beat until firm peaks form.
Add the whipped egg whites to the batter in several additions, gently folding with a spatula after each addition. Do not overmix.
Transfer the batter to the prepared baking dish and smooth the surface.
Wash and thoroughly dry the apricots. Cut them in half and remove the pits.
Arrange the apricot halves over the batter with the cut sides facing up, pressing them very lightly into the surface.
Bake for approximately 40 minutes, or until the top is golden and a toothpick inserted into a section without fruit comes out clean.
Allow the cake to cool in the pan for 15–20 minutes before slicing and serving.