In a small saucepan, melt the butter over medium heat. Continue to cook the butter, swirling the pan occasionally, for about 5 minutes, until the butter is fragrant and nutty golden brown solids have formed on the bottom of the pan. Remove the pan from the heat.
In a bowl, combine the flour, brown sugar, baking powder, 1/2 teaspoon salt and spices.
To toast the pecans, spread them out on a baking sheet and bake at 325 for 4-5 minutes, until hot and fragrant. Let cool for a few minutes, and then chop into small pieces. Add the toasted, chopped pecans to the flour mixture, reserving a few tablespoons of chopped pecans for garnish.
Stir the hot browned butter, scraping all the browned bits off the bottom of the saucepan, too, into the flour mixture. Add the eggs and vanilla, and stir the dough until combined. It will be fairly soft and sticky.
Cover and chill the dough for 1 - 1 1/2 hours.
Preheat the oven to 350 F, and line a large baking sheet with parchment paper.
Divide the chilled dough into 3 portions, about 10 ounces each. Dust the dough with a little flour to keep it from sticking, and shape each portion of dough into a slab (using your hands or a small rolling pin to press it out onto the baking sheet) measuring about 8 inches long, 3 inches wide, and 1/2 - 3/4 inch thick. Space the slabs of dough 4 inches apart on the baking sheet - they will spread somewhat while baking.
Brush the top of the dough with the egg white. Bake for 20 minutes, until the edges are just starting to lightly brown.
Remove the pan from the oven, and let cool for 10 minutes. Reduce the oven temperature to 325 F.
Use a large spatula or cake lifter to carefully lift the slabs of biscotti onto a cutting board. Use a sharp knife (I find that a chef's knife works best) to slice the slabs into individual biscotti about 3/4 inch thick. Place the biscotti back on the baking sheet, turned on their side with the cut side facing up.
Bake the biscotti for 10 minutes. Turn the biscotti over, so the other cut side is facing up, and bake for 10 more minutes. Cool completely.
Melt the white chocolate, using a double boiler, or in the microwave at 50% power, until smooth. Use a spoon to drizzle the white chocolate over the biscotti.
Before the chocolate sets, sprinkle it with the reserved chopped pecans and the rest of the sea salt. Let the chocolate harden completely before storing the biscotti in an airtight container.