Preheat oven to 350°. In a large bowl, cream butter and sugar until light and fluffy. Beat in egg and extracts. In another bowl, whisk flour, cinnamon, cream of tartar, baking soda, salt and nutmeg; gradually beat into creamed mixture. Stir in peaches.
Drop dough by tablespoonfuls 2 in. apart onto parchment-lined baking sheets. Bake until set, 14-16 minutes. Cool on pans 2 minutes. Remove to wire racks to cool. Store in an airtight container.