Preheat oven to 375°F. Line 2 baking sheets with parchment paper. Beat butter, brown sugar, molasses, and ½ cup of the granulated sugar with a stand mixer fitted with a paddle attachment on medium-high speed until smooth and fluffy, 2 to 3 minutes. Add egg and vanilla; beat until combined, about 30 seconds.

Whisk together flour, cinnamon, baking soda, ground ginger, cloves, nutmeg, and salt in a medium bowl until combined. Gradually add to butter mixture, beating on low speed until combined, about 2 minutes, stopping to scrape down sides of bowl as needed. Fold in crystallized ginger. Cover bowl with plastic wrap; refrigerate until chilled, at least 1 hour or up to overnight (8 hours).

Remove dough from refrigerator, and unwrap. Using a 2-inch cookie scoop (about 3 tablespoons), scoop cookie dough into rounds, and roll each into a smooth ball. (You should have about 30 balls.) Place remaining ½ cup granulated sugar on a plate. Roll balls in sugar to coat, and arrange 3 inches apart on prepared baking sheets. Bake in 2 batches in preheated oven until cookies have crackles and edges are slightly darker in color, 10 to 12 minutes per batch. Let cool on baking sheet 10 minutes. Transfer cookies to a wire rack, and cool completely, about 30 minutes.

Whisk together powdered sugar and 2 tablespoons lemon juice in a small bowl until combined and smooth; whisk in additional lemon juice 1 teaspoon at a time until desired consistency. Drizzle over cooled cookies; sprinkle with sparkling sugar or sanding sugar as desired.