Preheat oven to 375°F Line a cookie sheet with parchment or Silpat.
Combine flour, 1 1/2 tsp cinnamon, baking powder and salt. Set aside.
With a stand mixer or hand mixer, cream butter and 1 cup sugar until well combined, scraping down the bowl at least twice. Add the eggs, one at a time, mixing well to combine. Add the vanilla and mix until incorporated. Add the flour mixture by large spoonsful; mix only until it forms a crumbly dough. Do not overmix.
Turn dough out onto a lightly floured surface. With floured hands, knead dough once or twice to form a mass. Divide in two. Form each piece into a flat log, about 10" long, 4" wide and 1" high. Place both logs on cookie sheet.
In a bowl, whisk egg white and water. In another bowl, combine 1/4 c sugar and 1 tsp cinnamon. Brush both logs with egg white, then sprinkle evenly with cinnamon sugar.
Bake in preheated oven until small cracks form on top, about 25 minutes. Remove cookie sheet from oven, but do not turn oven off. Let cookie sheet stand about 10 minutes, then remove logs to a cutting board. With a bread knife, gently cut each log into 1/2" slices across the short dimension (so you get pieces that are about 4"x1"x1/2").
Place slices, cut side down, back on covered cookie sheet. Return to oven and bake 10 minutes. Flip biscotti over and bake another 10 minutes, until slightly browned. Immediately remove to a rack. When completely cool, transfer to airtight storage; they will keep at room temperature up to two weeks.