Preheat oven to 350º F (180º C). Line a cupcake pan with cupcake liners.
Combine milk and vinegar in a small bowl in a measuring cup or bowl. Set aside.
In a large bowl, sift flour, baking powder, baking soda, and salt. Set aside.
Using an electric hand mixer or a stand mixer, beat butter until creamy and pale on medium-high speed, about 2 - 3 minutes. Add the sugar and beat on high speed for 2 minutes until creamed and pale, scraping down the sides and up the bottom of the bowl with a rubber spatula as needed. Then add oil and vanilla; continue beating to combine, about 2 minutes.
Turn the mixer to low and add the flour mixture in three batches, alternating with milk mixture, beginning and ending with the flour mixture; beat until just combined, about 30 – 45 seconds. Do not overmix. Gently and slowly fold in the chopped Oreos. The batter will be slightly thick.
Fill cupcake liners about two-thirds full.
Bake for 20 - 22 minutes, or until a toothpick inserted in the center comes out clean. Cool pan on wire racks for 5 minutes, then remove cupcakes from pan, place back on the rack, and cool to room temperature before frosting, about 1 hour.
Frost cooled cupcakes with Oreo buttercream (you can find this recipe in my cookbook, page 158).