Preheat the oven to 350ºF (180ºC). Line three 7-inch (18-cm) round cake pans with parchment paper. Lightly grease with baking spray with flour.
Sift the flour, baking powder, baking soda, and salt together into a large bowl.
In a mixing bowl or liquid measuring cup, combine the milk and vinegar. Let the mixture rest for 5 to 8 minutes, until thickened and curdled.
Using an electric hand mixer or a stand mixer, beat the butter on medium-high speed until creamy, 3 to 4 minutes. Add the sugar and beat on high speed for 4 minutes, until creamed and pale, scraping down the sides and up the bottom of the bowl with a rubber spatula as needed. Then add the yogurt and vanilla; continue beating to combine, about 1 minute.
Turn the mixer to low and add the flour mixture in three batches, alternating with the milk mixture, beginning and ending with the flour mixture. Beat until just combined, 30 to 45 seconds. Do not overmix. Fold in the crushed Oreos.
Evenly divide the batter among the prepared pans.
Bake for 35 to 45 minutes, or until a toothpick inserted in the center comes out clean. Let the cakes cool in their pans on wire racks for 10 minutes, then remove from the pans. Allow the cakes to cool completely, right-side up, on wire racks before removing the parchment paper.