To prepare the brown butter, melt the butter in a medium-sized saucepan over medium heat. As the butter melts, whisk it frequently until foam begins to appear. Once you see the foam, whisk constantly until the butter begins to brown and it smells nutty and fragrant. Butter can go from brown to burnt very quickly so do not leave it unattended. Remove from the heat and allow to cool. After the butter has cooled, gently whisk it so that the brown bits evenly incorporate. Set aside 1/4 cup of the brown butter for the caramel and use the rest in the biscotti dough (approximately 7 fluid ounces).
In a large bowl, whisk together the flour, baking powder, cinnamon and 1/4 teaspoon salt. In a separate bowl, whisk together the sugar, vanilla, eggs and brown butter. Stir the wet ingredients into the dry until just incorporated. Cover and refrigerate for 30 minutes.
Preheat the oven to 350 degrees F. Line a large baking sheet with parchment paper. Remove the dough from the refrigerator and divide in quarters. On a lightly floured surface, roll each quarter into logs that are approximately 8x2 inches (two 16x2 logs will also work but I find the smaller logs easier to handle). Place the logs on the prepared baking sheet and bake for 25-30 minutes. Allow to cool for 15 minutes.
Once the logs have cooled, carefully move them to a cutting board and use a serrated knife to slice them on a bias into 3/4 inch slices. Arrange the biscotti on the same baking sheet cut side down and bake for another 20-25 minutes, or until golden.
To prepare the caramel sauce, mix the brown sugar, heavy cream, 1/4 cup browned butter (reserved from above) and 1/2 teaspoon kosher salt in a saucepan over medium-low heat. Bring to a simmer and the cook, whisking frequently, for 5 minutes or until the caramel sauce thickens. Turn off the heat, cool slightly, and then use a spoon or spatula to drizzle the caramel sauce over the biscotti.