Whisk flour, baking soda, and salt together and set aside. Combine butter and sugar in bowl of an electric mixer fitted with the paddle attachment. Beat on medium speed until light and fluffy. Add vanilla and egg and beat until smooth. Add flour mixture and beat until combined.
Remove bowl from mixer and using a wooden spoon, stir in dates, apricots, and pecans. Knead dough briefly until smooth, about 30 seconds. Divide dough in half. Form each half into a rectangular log, 2-by-1 1/2-by-7 1/2 inches. Wrap each log in parchment then in plastic wrap. Refrigerate or freeze until firm, about 4 hours in refrigerator or 2 hours in freezer. Dough can be made ahead up to this point and frozen for up to 3 months.
Heat oven to 375 degrees. Line baking sheets with Silpats (French nonstick baking mats) or parchment paper. Unwrap one log. Using a very sharp knife, slice dough into cut into 1/4- to 1/8-inch slices. Place on prepared sheets, 20 per sheet, about 1 inch apart. Bake until firm and barely browning around the edges, 10 to 12 minutes. Remove from oven and cool on sheets for about 3 minutes before transferring to wire racks to cool completely. When cool, drizzle with royal icing and let sit 1 hour to set icing.
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