Place gluten free flour, baking powder, salt and cinnamon in a large bowl; mix well.
In a separate bowl, mix vanilla extract, egg and vegetable oil until well combined.
Add dry flour mix to egg mixture and stir well with a wooden spoon. Don't worry, the pancake batter will be lumpy.
Pour in milk and mix well. Add more milk if needed until the batter is smooth. You might notice that the dry ingredients will absorb more liquid (if you set the mixture aside for a while) and the pancake batter will become thicker. Add more milk, 1 tablespoon at a time if necessary, to achieve the right consistency.
Heat oil in a pan over medium-low heat; pour 1 or 2 heaped tablespoons of the pancake batter into the pan to make a small pancake. Give it about 30 seconds, flip, repeat until cooked through. Repeat with remaining pancake batter.