Preheat oven to 350°F. Process freeze-dried strawberries and 4 tablespoons of the sugar in a food processor until mixture forms a fine powder, about 1 minute. Reserve ¼ cup powdered strawberry mixture for frosting. Set aside remaining ⅓ cup powdered strawberry mixture.

Beat butter and remaining ⅔ cup sugar in a stand mixer fitted with a paddle attachment on medium speed until light and fluffy, about 4 minutes. Add eggs, 1 at a time, beating until blended after each addition. Stir in vanilla, lemon zest, and lemon juice.

Whisk together flour, baking powder, salt, baking soda, and ⅓ cup powdered strawberry mixture in a medium bowl. Gradually add to butter mixture alternately with buttermilk, in 3 additions, beginning and ending with flour mixture. Beat just until blended after each addition. Gently fold in chopped strawberries. Place 12 paper liners in a 12-well standard-size muffin pan. Divide batter evenly among wells (about ⅓ cup each).

Bake in preheated oven until a wooden pick inserted in centers comes out clean, 22 to 24 minutes. Cool in pans on wire racks 5 minutes. Remove from pans to wire racks, and cool completely, about 30 minutes. Prepare the Frosting

Beat cream cheese and butter in a stand mixer fitted with a paddle attachment on medium-high until pale and fluffy, about 3 minutes. Reduce speed to low, and gradually beat in vanilla, powdered sugar, and reserved ¼ cup powdered strawberry mixture. Increase speed to medium-high, and beat until well combined and fluffy, about 1 minute. Transfer frosting to a large piping bag fitted with a ½-inch closed star tip. Pipe frosting evenly over cupcakes. Garnish with strawberry halves. Serve immediately, or store, covered, in refrigerator for up to 3 days.