First, chop the peaches and set them aside. In a large mixing bowl, whisk together the flour and sugar. Make a well in the center.
In a separate medium mxing bowl, combine the milk, melted butter, egg, and vanilla extract and mix until they are well combined.
Pour the wet ingredients into the well in the dry ingredients. Using the whisk, gently fold in the wet ingredients, making sure to mix in all of the flour on the sides and bottom of the bowl. Make sure the ingredients are well combined but don't overmix.
With a rubber spatula, gently fold in the peaches.
Preheat a large nonstick skillet or griddle on medium heat. Use a brush to coat the skillet with oil or butter before each batch of pancakes. First, cook one test pancake to make sure the temperature is right. Pour about 1/4 a cup of batter onto the skillet and let it cook until large bubbles have formed and the underside is medium brown, which should take 2-3 minutes. Flip and cook the other side for about 1-2 minutes. Adjust the heat if necessary and cook the rest of the pancakes 2 or 3 at a time, depending on the size of your skillet.