In a medium bowl, whisk together flour, baking powder, brown sugar, and salt. Create a well in center of bowl and pour milk, egg, 2 tablespoons vegetable oil, and vanilla extract into well. Whisk into dry ingredients until batter is smooth and transfer batter to a squeeze bottle with a 1/4-inch opening at top (or to a wet measuring cup).

Holding squeeze bottle 1 inch from oil, carefully squeeze batter into oil in a spiderweb pattern, starting from center, using about 1/4 cup batter per cake. Fry cake, flipping once, until golden brown on both sides, 3-4 minutes total. Transfer to a paper towel-lined plate to drain. Return oil to 350 degrees before frying next cake.