Begin by heating the oil in a deep, heavy-bottomed saucepan over medium heat. Allow the oil to gradually reach a temperature suitable for frying while preparing the batter.
In a large mixing bowl, whisk together the yogurt or sour cream, milk, egg, and melted butter until smooth and well combined.
In a separate bowl, blend the flour, baking powder, cinnamon, sugar, and salt. Gradually incorporate the dry ingredients into the wet mixture, stirring only until just mixed. Avoid overmixing to maintain a light texture.
Carefully fold in the chopped peaches along with any juice collected, ensuring the fruit is evenly dispersed throughout the batter. The consistency should resemble that of cake batter—neither overly thick nor too runny.
Using a spoon or small scoop, drop about 1 to 2 tablespoons of batter into the hot oil. Fry the fritters in small batches, turning as needed, until both sides are an even golden brown—approximately 2 to 3 minutes per side. Once fried, transfer them to paper towels to remove excess oil, then place them on a wire rack to cool slightly.
To prepare the glaze, whisk together the powdered sugar, vanilla extract (or reserved peach juice), and milk in a bowl until completely smooth and free of lumps. Drizzle the glaze generously over the slightly cooled fritters.