For fig filling: In a medium saucepan, combine all ingredients and cook over medium-low heat for about 1 hour or until liquid has reduced to the point that it's covering figs about halfway. Transfer to a food processor and blend until smooth. Transfer to a bowl, cover, and refrigerate until chilled. Makes about 1 cup filling, which can be kept in fridge for up to 1 month.
For cookies: Use a spice grinder or mortar & pestle to grind 1/4 cup of the sesame seeds with 1 Tbsp. of sugar. Whisk mixture together with flour, baking powder, and salt. In the bowl of an electric mixer, beat butter and remaining 5 Tbsp. sugar together until creamy. Add egg and vanilla and mix until combined. Set bowl of dough in refrigerator to chill for about 10 min. Meanwhile, line two baking sheets with parchment paper and preheat oven to 350 degrees. Use a tablespoon to scoop dough about 1 inch apart onto baking sheets. Bake until just set, about 6 to 8 min. Remove sheets from oven and use a teaspoon to press a small well into the middle of each cookie. Fill each well with about 1/2 tsp. fig filling. Return cookies to oven to bake for 6 to 8 more min. or until golden brown around the edges. Remove from oven and sprinkle each cookie with sesame seeds. Let cool completely on baking sheets before serving.