1. Preheat oven to 450°.
2. Weigh or lightly spoon flours into dry measuring cups; level with a knife. Combine flours, brown sugar, baking powder, salt, and baking soda in a large bowl, stirring with a whisk to combine.
3. Place butter in a medium microwave-safe bowl. Microwave at HIGH for 1 minute or until melted. Add cold buttermilk, stirring until butter forms small lumps. Add oil and vanilla, stirring to combine.
4. Add buttermilk mixture to flour mixture, stirring just until incorporated (do not overmix) and batter pulls away from sides of bowl. (Batter will be very wet.)
5. Drop batter into 12 mounds (about 2 heaping tablespoonfuls each) onto a baking sheet lined with parchment paper. Bake at 450° for 12 minutes or until golden.
6. While biscuits bake, slice strawberries. Combine strawberries, 2 tablespoons sugar, and lemon juice; toss to coat. Set aside.
7. Place yogurt, cream, and remaining 2 tablespoons sugar in a bowl; beat with a mixer at medium speed until soft peaks form. Split biscuits; fill with berry mixture and cream.