Heat oven to 350°F. Line two baking sheets with parchment.
Whisk flour, baking powder, baking soda, salt, and cinnamon in a bowl.
Beat softened butter, granulated sugar, and brown sugar until very fluffy, about 2 minutes.
Mix in the egg and 2 teaspoons vanilla until smooth and creamy.
Add dry ingredients in two additions, alternating with buttermilk, beginning and ending with dry. Mix just to combine.
Fold in the finely diced, well-dried peaches.
Scoop 2-tablespoon mounds onto sheets, spacing about 2 inches apart.
Bake 10–12 minutes until set and lightly golden at the edges. Cool on sheets 5 minutes, then transfer to a rack to cool fully.
Beat cream cheese and softened butter until perfectly smooth.
Gradually mix in powdered sugar, 1 teaspoon vanilla, and heavy cream until fluffy. Chill 10 minutes to firm slightly.
Pair cookies by size. Pipe or spread filling on the flat side of half. Swirl in a little peach jam if using.
Cap with remaining cookies to make sandwiches. Chill 15 minutes to set before serving.