Line two baking sheets with parchment. Place the freeze-dried strawberries in a zip-top bag and crush with a rolling pin. Measure out 1 tablespoon of the finest powder and set it aside; keep the rest in small pieces.
Whisk the flour, baking soda, salt, and the reserved 1 tablespoon strawberry powder in a medium bowl until evenly combined. Set aside.
Make the cheesecake center: In a small bowl, beat the cream cheese, powdered sugar, and 1/2 teaspoon vanilla until smooth. Scoop 18 small mounds (about 1 1/2 teaspoons each) onto a parchment-lined plate and freeze until firm, 25–30 minutes.
For the cookie dough, beat the butter, brown sugar, and granulated sugar on medium-high until fluffy, 2–3 minutes. Scrape the bowl. Mix in the egg, egg yolk, 1 1/2 teaspoons vanilla, and lemon zest (if using) just until combined.
Add the dry ingredients to the bowl and mix on low just until a few flour streaks remain. Fold in the white chocolate chips and the remaining crushed strawberry pieces until distributed.
Portion dough into 1/4-cup scoops. Working one at a time, flatten a portion in your palm, place a frozen cheesecake mound in the center, and pinch the dough closed to seal. Roll gently into a ball and place on the prepared sheets, spacing 2 1/2 inches apart.
Chill the tray of stuffed dough balls for 25–30 minutes. While they chill, preheat the oven to 350°F with a rack in the center.
Bake 13–15 minutes, rotating the pan once, until the edges are lightly golden and the centers look set but still soft. Do not overbake—the middles will finish setting as they cool.
Cool on the sheet for 10 minutes, then transfer to a rack to cool completely before storing.