In a large bowl or the bowl of a stand mixer with paddle attachment combine butter, sugar, brown sugar, and molasses. Beat together on medium speed until light and fluffy, about 5-6 minutes. Scrape down side of the bowl once or twice during the process.
On low speed add egg and beat until combined. Add sourdough starter and beat until combined. Scrape down side of bowl and mix again until evenly combined.
In a medium bowl mix together flour, baking soda, ginger, cinnamon, cloves, and salt.
Gradually mix dry ingredient mixture into wet ingredient mixture on low speed. Mix until just combined. Scrape down side of bowl and mix again until evenly combined.
Cover dough and chill for at least 30 minutes.
Preheat oven to 350°. Line two large sheet pans with parchment paper.
Place turbinado, Demerara, or granulated sugar in a shallow bowl. Scoop tablespoon sized balls of dough and roll in sugar. Place on sheet pans about 2-inch apart (I usually fit 12 on a standard sized 13x18-inch half sheet pan).
Bake for 12-14 minutes or until they are a deep and even golden-brown color with significant cracking on top. Allow to rest on baking sheets for 2-3 minutes then transfer to a wire rack to cool completely.