Preheat the oven to 350°F (177°C). Grease three 9-inch cake pans and line the bottoms with parchment paper.
In a large bowl, whisk together the flour, baking powder, baking soda, and salt. Set aside.
In a stand mixer bowl fitted with a whisk attachment (or use an electric hand mixer), cream the softened butter and sugar on medium speed until light and fluffy—about 2–3 minutes.
Add the oil and mix again until smooth. Add the eggs, one at a time, mixing well after each. Scrape down the sides of the bowl as needed. Add the sourdough discard, and vanilla extract.
With the mixer on low, add the dry ingredients to the butter mixture just until combined. With the mixer still on low add the milk. Mix until just combined.
Divide the cake batter evenly between the 3 prepared pans (use a scale for equal parts if desired).
Bake for 25-27 minutes or until a cake tester comes out clean and the tops are golden brown.
Let the cakes cool in the pans until completely cool, then carefully turn them out onto a wire rack.