In a large bowl, stir together sugar, butter, and vanilla extract. Add eggs and mix to combine. Add the flour mixture 1/3 at a time to the butter/egg mixture and stir until just combined.
1 cup Sugar, 1/2 cup Butter, 1 1/2 tsp Vanilla extract, 3 Eggs
Dampen your hands and shape dough into 2 4"x10" logs on a parchment-lined baking sheet. Beat the egg with the teaspoon of water. Brush on the egg wash.
Bake about 30-35 minutes until lightly golden. Transfer the baking pan to a wire rack and cool for 5-10 minutes, or cool enough to handle.
Using a very sharp, serrated knife, cut the logs into 3/4-inch slices and arrange on the baking sheet, cut side down.
Bake biscotti again for 5-10 minutes. Remove from the oven, turn each piece over to the other cut side, and bake another 5-10 minutes until golden and crispy. Watch them closely if your oven runs hot; they will burn quickly. Transfer to a wire rack to cool completely.
Melt the chocolate chips in separate zip-lock bags in the microwave at 30-second intervals until almost completely melted. Squish it around in the bag until smooth, cut a tiny bit off one corner, and drizzle. Allow the chocolate to set up completely before storing.
Notes
Keep in an airtight container at room temperature for up to 2 weeks or freeze for up to 3 months.