Place 1 cup milk in a microwave-safe bowl; heat in microwave until bubbling, about 2 minutes. Stir in the oats.
Mix flour, brown sugar, baking powder, salt, and cinnamon together in a bowl. Whisk eggs, 2 teaspoons melted butter, and vanilla extract together in a separate bowl; add 1/4 cup milk. Mix oat mixture and egg mixture with flour mixture; stir to combine.
Heat 2 teaspoons butter on a griddle or skillet over medium heat; pour about 1/3 cup batter into the hot butter. Cook pancakes until bubbles appear on top layer, about 5 minutes. Flip and cook other side until evenly browned, about 5 more minutes.