Whisk flour, cardamom, and salt in medium bowl to blend. Using electric mixer, beat butter in large bowl until creamy. Gradually add sugar; beat until light and fluffy. Beat in finely grated orange peel and vanilla. Add egg; beat to blend. Add flour mixture in three parts, beating on low speed until just blended between additions. Refrigerate until firm enough to shape.
Divide dough in half. Form halves into ball, flatten into disks and wrap tightly in plastic. Chill until firm 45 minutes to overnight.
Position racks in top and bottom third of oven; preheat to 350°F Line 2 large baking sheets with parchment paper. Roll out 1 dough disk on lightly floured surface to generous 1/8" thickness. Cut out cookies and transfer to prepared baking sheets, spacing 1" apart. Gather dough scraps into ball. Flatten, cover, and freeze dough until firm enough to roll out again, about 10 minutes.
Bake cookies until light golden brown, about 16 minutes, rotating sheets halfway through baking for even cooking. Carefully slide parchment paper with cookies onto racks to cook completely.