Prepare the chopped strawberry mixture. Mix the chopped strawberries, lemon juice, and sugar in a small bowl. Stir well and set aside to let the flavors blend.
Make the cookie dough. In a large mixing bowl, beat the shortening (or unsalted butter), granulated sugar, egg, lemon juice, and lemon zest together until the mixture is light and fluffy. In a separate bowl, whisk together the flour, baking soda, and salt.
Combine the ingredients to the dough. Add the dry ingredients to the wet mixture slowly, mixing at low speed until just combined. The dough will look crumbly. Remove the strawberries from the earlier mixture, and only add the berries (not the liquid) to the dough. The moisture from the berries will help the dough stick together. Cover the dough and put it in the fridge for 30 minutes.
Bake the cookies. Remember to heat up the oven to 350°F (175°C) while the dough chills. Take 1-2 tablespoons of dough for each cookie and put them on a baking sheet lined with parchment paper. Then, bake them for 10-12 minutes, depending on the cookie size, until the edges are firm and the tops have a few cracks. After that, take them out from the oven and move the cookies to a wire rack to cool down completely.
Make the icing. Mix the powdered sugar with the reserved strawberry juice in a medium bowl until smooth. The icing should be thick and spreadable but not pourable. Add more lemon juice or powdered sugar if needed. Test the icing on one cookie to make sure it stays on before frosting the rest. After the cookies are completely cool, spread the icing over each cookie. You can keep the cookies at room temperature for one day or refrigerate them for longer storage.