Preheat oven to 350°F. Line sheet pan with parchment paper.
In bowl of stand mixer fitted with paddle attachment, or in large bowl with hand-held mixer, beat butter and brown sugar on high speed until light in color and creamy. Mix in egg, followed by molasses.
In large bowl, stir together flour, baking soda, cinnamon, cloves and ginger. Add flour mixture to butter mixture all at once and mix on low until incorporated.
Divide dough in half and shape each piece into log about 4 inches wide and space evenly on the prepared sheet pan. Flatten each log to about 2 inches tall. Sprinkle generously with granulated sugar.
Bake until edges are golden and surface is cracked, 20-25 minutes. Remove from oven and let cool on pan 20 minutes.
Use serrated knife to slice logs into 1-inch thick slices. Place slices cut side-up on pan lined with parchment paper and bake additional 12-14 minutes until edges are brown. Let cool completely.
In small heat-proof bowl, melt white chocolate in microwave at 60% power for 1 minute. Stir until smooth. If not completely melted, continue to microwave in 10 second intervals, being careful not to scorch.
Drizzle melted chocolate over cooled cookies. Let sit 15 minutes before serving or storing. Store biscotti in airtight container at room temperature up to 1 week.