Place the butter, brown sugar, and 1 tablespoon of the lavender sugar in the bowl of an electric mixer. Mix on medium for 30 seconds and then cream on medium-high for about 2 minutes.
Add the egg and vanilla. Mix on low until blended. Scrape the sides of the bowl as needed.
Sprinkle in the salt. With the mixer on low, add the flour a little at a time. Increase the speed to medium and mix just until all the flour is incorporated into the dough. Refrigerate the dough for 30 minutes.
Preheat the oven to 350 degrees F.
Divide the dough into 20 balls. Divide each ball in two and flatten each piece into a round disk. Place a slice of fig on one flattened disk. Place the other disk on top and form the dough into a disk around the fig slice. Place the cookies on an ungreased baking sheet, or a baking sheet covered in a silicone baking mat. Sprinkle with some of the remaining lavender sugar.
Bake for 13 to 15 minutes, until the edges of the cookies brown and centers are firm. Remove from oven and let rest for 2 minutes. Transfer to a cooling rack and allow to cool completely before serving.
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