Make the filling: In a small bowl, stir together peach preserves, diced peach, cinnamon, nutmeg, and lemon juice. Set aside. The diced peach adds texture; don't skip it.

In a medium bowl, whisk together flour, baking soda, and salt. Set aside.

In a large bowl, beat butter, granulated sugar, and brown sugar with an electric mixer on medium speed until light and fluffy, about 3 minutes. Scrape the bowl.

Add eggs one at a time, beating well after each. Beat in vanilla and milk until combined.

Gradually add the flour mixture, mixing on low just until no streaks of flour remain. Do not overmix. The dough will be soft but not sticky.

Cover and chill the dough for at least 30 minutes. This prevents spreading and makes the cookies puffier.

Preheat oven to 350°F (175°C). Line two baking sheets with parchment paper.

Scoop 1.5-tablespoon portions of dough (a small cookie scoop works perfectly). Roll into balls and place 3 inches apart on the prepared sheets.

Use your thumb or the back of a 1/2-teaspoon measuring spoon to make a deep well in the center of each dough ball. The well should be about 1/2 inch deep and wide enough to hold a generous teaspoon of filling.

Spoon about 1 heaping teaspoon of the peach filling into each well. Do not overfill—the filling will bubble as it bakes.

Bake one sheet at a time for 11–13 minutes, until the edges are set and lightly golden, and the tops look dry around the edges but still slightly soft in the center. The filling will be bubbly.

Let cookies cool on the baking sheet for 5 minutes—this firms them up so they don't fall apart when moved. Then transfer to a wire rack to cool completely. The filling will thicken as it cools.