Preheat the oven to 375°F. Line a few baking sheets with parchment paper.
Whisk together the flour, baking powder, and salt in a small bowl to combine.
Beat the sugar and butter in the bowl of an electric mixer on medium speed for about 2 minutes, until light and fluffy. Add the egg, sour cream, lemon zest, and lemon juice and mix until well combined. With the mixer on low speed, add the flour mixture in two additions, blending until just incorporated. Stir in the pistachios.
Using a small ice cream scoop, drop the dough onto the prepared baking sheets, spacing the cookies 1-inch apart. Bake for about 15 minutes, until firm to the touch and lightly browned around the edges. Remove from the oven and let cool for 5 minutes. Transfer the cookies to a wire rack and let cool completely.
Combine the sugar and lemon juice in a small saucepan over medium heat, stirring until the sugar dissolves. Spoon some syrup over each cookie and sprinkle the pistachios over the top.