Whisk flour, baking powder, salt, ground ginger, cinnamon, nutmeg, and cayenne together in a bowl until thoroughly combined.

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Combine butter, brown sugar, and molasses in a bowl; mix with a spatula until creamy. Whisk in eggs, one at a time, until smooth. Add vanilla and chopped ginger. Mix and mash with the whisk until ginger is evenly distributed and coated with the butter-sugar mixture, about 1 minute.

Mix in 1/2 of the dry ingredients with a spatula until incorporated. Mix in remaining dry ingredients until dough is very wet and sticky.

Transfer dough to a long piece of plastic wrap, using a buttered knife to help if necessary. Wrap plastic around the dough and shape with your hands into a uniform log, about 12 to 14 inches long by 2 1/2 inches wide. Place in the refrigerator to chill for at least 30 minutes.

Preheat the oven to 350 degrees F (175 degrees C). Line a baking sheet with a silicone baking mat.

Remove dough from the refrigerator, unwrap, and place on the prepared baking sheet. Wet your fingers and smooth out any edges.

Bake in the center of the preheated oven until a toothpick inserted into the center comes out clean, about 30 minutes.

Remove loaf from the oven, cover with a clean towel, and let sit for 20 minutes. Reduce oven temperature to 325 degrees F (165 degrees C).

Uncover the loaf and brush a little water around the bottom edge to soften. Transfer to a cutting board.

Use a serrated knife to cut the loaf into 1/2-inch slices. Cut at an angle for longer biscotti, or straight across for shorter ones. Transfer slices back to the baking sheet.

Return to the oven and bake for 20 minutes. Flip biscotti and cook until they are dry and crispy all the way through, 10 to 15 minutes longer. Transfer to a cooling rack and let cool to room temperature.

Place melted white chocolate into a plastic bag. Snip off a corner and pipe over biscotti. Let sit until chocolate has cooled and hardened, about 30 minutes.