In a heavy duty saucepan, combine the sugar, water, and lemon juice. Bring the mixture to a boil over medium-high heat. Reduce to a simmer over medium-low heat and add the lemon slices in a single layer. Gently flip the lemon slices once or twice during the cooking process. Simmer for 15 minutes.
Transfer the lemon slices to a wire cooling rack or large piece of wax paper and allow them to cool completely.
Cookie
Preheat the oven to 350 F and line three baking sheets with parchment paper. Using a stand mixer fitted with the paddle attachment (or a large mixing bowl with a hand mixer), combine the butter, sugar, egg and lemon extract and beat on medium speed until creamy and free of lumps.
Place the freeze dried raspberries in a coffee grinder or small food processor and grind into a powder.
In a separate bowl, combine the raspberry powder, flour, baking powder, and salt. Mix well. Gradually add to the butter-sugar mixture and beat on medium speed until just combined. Add the lemon zest and lemon juice and mix until just combined. Fold in the fresh raspberries.
Using a medium cookie scoop, drop the dough by 1 1/2 tablespoons onto the prepared baking sheets. Bake in the preheated oven for 9-11 minutes. Let cool.
Glaze
Using a stand mixer fitted with the whisk attachment (or a large mixing bowl with a hand mixer), combine the confectioner’s sugar, lemon extract, lemon juice, and milk. Beat on medium speed until free of lumps.
Assembly
Generously drizzle the glaze over the cookies. Top each cookie with a half of a cooled candied lemon slice.
Notes
Change up the flavor of your pink lemonade cookies by using freeze-dried strawberries instead of raspberries.
For another decoration option, sprinkle the cookies with granulated sugar before baking. You could also dust them with powdered sugar after they've been baked!