Heat oven to 350°F. Line two baking sheets with parchment.
Toss diced peaches with lemon juice, cornstarch, and 1 tbsp sugar. Blot excess moisture with a paper towel.
Whisk flour, baking powder, and salt in a bowl. Set aside.
Cream butter, granulated sugar, and brown sugar until light and fluffy, about 2 minutes.
Beat in vanilla. Stream in cold heavy cream and mix until smooth and slightly thickened.
Add dry ingredients to the bowl. Mix on low just until a soft dough forms.
Fold in peaches gently. Chill dough 10–15 minutes to firm up for easier scooping.
Scoop heaping tablespoons of dough onto sheets, spacing 2 inches apart. Sprinkle tops with turbinado sugar.
Bake 12–14 minutes, until edges set and lightly golden. Cool on the pan 5 minutes, then transfer to a rack.
Whisk powdered sugar with lemon juice and just enough milk for a thick drizzle. Glaze cooled cookies.