Combine gluten-free flour, baking powder, salt, and cinnamon in a large bowl; mix well.
Mix egg, vegetable oil, and vanilla extract together in a second bowl until well combined. Stir flour mixture into egg mixture with a wooden spoon until a lumpy batter forms. Pour in milk and mix until a smooth batter forms. Set batter aside to thicken, 5 to 10 minutes. Add more milk, 1 tablespoon at a time, if batter is too thick.
Heat oil in a griddle over medium-low heat. Drop 1 to 2 heaped tablespoonfuls onto the griddle for each pancake. Cook until bubbles form and the edges are dry, 3 to 4 minutes. Flip and cook until browned on the other side, 2 to 3 minutes. Repeat with remaining batter.
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