Melt butter in a small saucepan over low heat. Cook until milk solids stop crackling and turn amber (about 5 minutes), stirring occasionally. Transfer butter mixture to a small bowl, scraping pan to include milk solids. Cover and cool butter mixture in the refrigerator for 20 minutes or until soft and congealed but not firm.
Lightly spoon flour into dry measuring cups, and level with a knife. Combine flour and next 5 ingredients (through nutmeg) in a bowl, stirring with a whisk.
Combine chilled butter mixture and granulated sugar in a large bowl; beat with a mixer at medium speed until well blended (about 3 minutes). Add syrup, flavoring, and 1 egg white to butter mixture; beat at low speed 2 minutes or until well blended. Add flour mixture to butter mixture; beat on low speed until blended. Divide dough in half. Shape each portion into a ball; wrap in plastic wrap. Chill 1 hour or until firm.
Preheat oven to 350°.
Place walnuts and turbinado sugar in a food processor; pulse 15 times or until mixture is coarsely ground. Place remaining 1 egg white in another small bowl; stir with a whisk.
Working with one portion of the dough at a time (keep remaining dough chilled until use), roll dough to a 1/8-inch thickness on a floured surface, and cut with a 2 1/2-inch round or decorative cutter. Place 24 cookies, evenly spaced, on a baking sheet coated with cooking spray. Gently brush tops of cookies with egg white; sprinkle evenly with half of walnut mixture. Bake at 350° for 12 minutes or until pale brown. Remove cookies from pan; cool completely on wire racks. Repeat procedure with remaining dough, egg white, and walnut mixture.