Preheat the oven to 450 degrees F (230 degrees C). Grease a baking sheet.
Mix cream cheese and oil in a large bowl. Add all-purpose and whole wheat flours, baking powder, sugar, salt, and baking soda. Mix until large crumbs form. Add buttermilk and mix until dough comes together in a ball. Add up to 1/4 cup more buttermilk if needed.
Turn dough out onto a work surface and knead until smooth, about 5 minutes. Roll out to 1/2 inch thickness. Use a round cookie cutter to cut out 12 biscuits. Place biscuits on the oiled baking sheet.
Bake in the preheated oven until golden brown, 10 to 12 minutes. Immediately remove from the baking sheet. Serve warm.
