Soak oats in 3/4 cup milk for 10 minutes.
Meanwhile, whisk together flour, baking powder, baking soda, cinnamon, and salt in a large bowl.
Stir egg, butter, brown sugar, remaining 3/4 cup plus 2 tablespoons milk, and oat mixture into dry ingredients until just combined. If the batter is too thin, add flour.
Heat a griddle over medium heat until hot and lightly brush with oil. Working in batches, pour 1/4 cup batter per pancake onto griddle and cook until bubbles appear on surface and undersides are golden-brown, about 1 minute. Drop blueberries into pancakes. Flip with a spatula and cook the other side, about 1 minute more. (Lightly oil griddle between batches.)