Preheat oven to 350°F. Line heavy large baking sheet with parchment paper. Place 1 1/2 cups almonds in large bowl and sift 1/2 cup powdered sugar over; toss to coat. Whisk 1 egg white to blend in small bowl. Add 1 tablespoon egg white to nuts and toss to blend. Spread mixture in even layer over prepared baking sheet. Bake until almonds are golden and coating is dry, about 10 minutes. Cool completely. Transfer candied almonds to processor. Using on/off turns, process until almost all almonds are coarsely chopped and some fine crumbs form. Transfer chopped candied almonds to small bowl. Line 2 large baking sheets with parchment paper. Finely chop remaining 1 1/2 cups slivered almonds in processor. Add flour, baking soda, and salt. Blend until nuts are very finely ground. Using electric mixer, beat butter, 3/4 cup sugar, and remaining 3/4 cup powdered sugar in large bowl until well blended. Add remaining 2 egg whites, amaretto, and vanilla. Beat until well blended. Gradually mix in flour-almond mixture. Stir in chopped candied almonds. Working in 2 batches and using small ice cream scoop or tablespoon, spoon 2 level tablespoonfuls dough in mound for each cookie, forming about 12 mounds on each sheet and spacing 2 inches apart. Bake until cookies are golden brown, about 14 minutes. Cool slightly on baking sheets. Transfer cookies to rack and cool completely. (Can be made 3 days ahead. Store airtight at room temperature.)