Preheat oven to 325°. Toast hazelnuts on a rimmed baking sheet until light golden brown and skins start to split, 8 to 10 minutes. Rub warm nuts in a kitchen towel to remove most of skins (it's okay if some stick). Let cool and chop coarsely.
Using a stand mixer with the paddle attachment (or a handheld mixer with beaters; cookies will be slightly crumblier), beat butter, granulated sugar, and salt on medium speed until evenly mixed and fluffy, 4 to 5 minutes.
In a small bowl, whisk together vanilla-bean paste and cognac. Scrape into butter mixture and mix to incorporate.
On low speed, gradually mix in flour, scraping bowl as needed. Add hazelnuts and mix 30 seconds more.
Immediately scoop cookies into 1 1/2-tbsp. (25 g) portions and roll into balls. Chill, covered, at least 4 hours and up to 2 days.
Preheat oven to 325° with racks in upper and lower thirds. Stack a baking sheet on a second sheet (to help cookies get evenly golden brown). Repeat with 2 more baking sheets. Line top sheet in each stack with parchment paper and set chilled cookie portions on parchment about 1 in. apart.
Bake cookies 20 minutes. Switch positions of pans and rotate each 180°; then bake until medium golden brown, almost like biscotti, 25 to 30 minutes more. Slide parchment with cookies onto cooling racks and let cool completely.
Put powdered sugar in a large, wide bowl, then roll cookies in powdered sugar. Repeat, lightly pressing sugar onto cookies to form a thick, fluffy coat.
*Pura recommends weighing ingredients on a kitchen scale, since it's more precise. Find vanilla-bean paste at surlatable.com.
Make ahead: Through step 5, up to 2 days, chilled airtight, or 2 months, frozen (thaw overnight in fridge before proceeding). Baked cookies, 2 to 3 days, airtight at room temperature, or up to 2 months, frozen.