Peel the peaches if desired, cut them in half, and remove the pits. Finely dice two peaches into pea-sized pieces, spread on a paper towel, and pat dry. Slice the third peach into thin wedges for topping and set aside.
In a large mixing bowl, whisk together the flour, sugar, baking powder, baking soda, salt, cinnamon, and nutmeg for 20 to 30 seconds. Make a small well in the center.
In a second bowl, whisk the eggs until blended. Add the buttermilk, cooled melted butter, and vanilla extract, then whisk until smooth.
Pour the wet ingredients into the well in the dry ingredients. Stir gently with a spatula just until no dry streaks remain, stopping while the batter still looks slightly lumpy.
Add the patted-dry diced peaches and fold them in with two or three gentle strokes. Let the batter rest for 5 to 10 minutes while the griddle heats.
Set a nonstick griddle or wide skillet over medium heat. Add a small pat of butter and swirl to coat. Test the heat by flicking on a few drops of water; they should sizzle and dance.
Pour a scant 1/4 cup of batter per pancake onto the griddle, leaving space between them. Cook until bubbles form across the surface and the edges look set, about 2 to 3 minutes. Flip once and cook the second side for another 1 to 2 minutes, until golden brown and cooked through.
Transfer the finished pancakes to a warm plate as you go. Stack them up, add the reserved sliced peaches on top if using, and finish with a pour of maple syrup. Serve right away.