In heavy, medium saucepan, over medium-high heat, bring chopped pecans and butter to a simmer. Cook, stirring frequently, 4 to 5 minutes, or until pecans and butter are nicely browned; be careful not to burn mixture. Into medium metal bowl, strain mixture through sieve; let stand until butter drains into bowl. Place butter in freezer, stirring once or twice, until firmed up but not hard, about 10 minutes. Set pecans aside. In food processor, combine chilled browned butter, oil, flour and confectioners sugar. Process, using on-off pulses, until well blended, scraping down sides of bowl occasionally. Scrape mixture into large bowl; stir in 3/4 C. reserved chopped pecans. In food processor, combine remaining 3/4 C. chopped pecans and the brown sugar; process until pecans are finely ground and mixture is crumbly, scraping down sides of bowl once or twice. Add egg and egg yolk, 1 1/2 tbsp water, the vanilla and salt; process until well blended, about 30 seconds. Scrape into bowl with butter flour mixture; stir until just blended. Knead with hands until blended and smooth. Divide dough in half; place each half between sheets of waxed paper. Roll out to generous 1/4" thickness. Stack rolled dough on baking sheet; freeze 15 minutes or until chilled and slightly firmed up but not hard. Place oven rack in upper third of oven. Preheat oven to 350. With spray, coat several baking sheets. Working with half dough at a time, peel away top sheet of waxed paper, then pat it back into place. Turn dough over, then peel off top sheet. Using floured fluted or plain round 2 1/4 to 2 1/2" cutter, cut out cookies. (If dough softens too much, refreeze until firm). Using spatula, transfer cookies to prepared baking sheets, spacing 1" apart. Let cookies stand a few minutes to soften slightly. Gently press pecan half into center of each. Re-roll any dough scraps; cut out. Bake cookies, 1 sheet at a time, 10 to 14 minutes, reversing baking sheet from front to back halfway through baking, or until barely colored on top and just slightly darker at the edges. Let cookies cool on baking sheet on wire rack 2 minutes, transfer cookies to wire rack; let cool completely. Store in airtight container up to weeks or freeze up to 1 month.