Preheat the oven to 350 degrees Fahrenheit and line 2-3 large baking sheets with parchment paper to prevent sticking.

In a large bowl or stand mixer, combine the granulated sugar and lemon zest. Rub the zest into the sugar with your fingers for 30-60 seconds to release the fragrant oils, enhancing the lemon flavor in the cookies.

Add the softened butter to the lemon-sugar mixture and beat until fluffy, about 3 minutes. Scrape down the sides of the bowl to ensure even mixing.

Beat in the egg, 2 tablespoons of lemon juice, and vanilla extract until well combined and smooth, creating a flavorful and aromatic base for the dough.

In a separate bowl, whisk together the all-purpose flour, salt, and baking soda to evenly distribute the leavening agents and salt.

With the mixer on low speed, slowly add the flour mixture to the butter mixture and mix until just combined. Avoid overmixing to keep the dough tender; the dough will be somewhat soft.

Using a 1-ounce cookie scoop (or 2 tablespoons), portion out the dough and roll each scoop into a ball with your hands. Place the dough balls on the prepared baking sheets, spacing them at least 2 inches apart to allow room for spreading.

Bake the cookies at 350 degrees Fahrenheit for 10-13 minutes until the edges are just set and the bottoms turn a light golden brown. This ensures they are tender with a slight crisp on the edges.

Allow the cookies to cool on the baking sheets for 10 minutes before transferring them to a wire rack to cool completely, preventing breakage and allowing them to firm up.

Add the freeze-dried raspberries to a blender or food processor and pulse until they are fine and powdery, with some small bits remaining for texture.

Beat the softened cream cheese and butter together in a mixing bowl until light and fluffy, about 3-4 minutes, creating a smooth, creamy base for the frosting.

Beat the powdered freeze-dried raspberries into the cream cheese mixture, evenly distributing the raspberry flavor throughout the frosting.

With the mixer on low, gradually beat in the powdered sugar one cup at a time until the frosting becomes smooth and thick. Scrape down the sides and bottom of the bowl as needed. Finally, beat in the milk or cream to achieve the desired consistency for spreading or piping.

Use a round piping tip to pipe the frosting onto the cooled cookies, or spread it with a palette knife. Garnish the frosted cookies with crushed freeze-dried raspberries for an extra burst of flavor and decorative touch.