Preheat oven to 350 degrees. Whisk together flour, baking soda, cream of tartar, and salt. Beat together butter, sugar, and orange zest with a mixer on medium-high speed until light and fluffy, about 3 minutes. Add vanilla, then egg whites, and beat on medium-high speed until well combined, about 3 minutes. Stir in flour mixture, then nuts and oats.
Transfer mixture to a pastry bag fitted with a 1/2-inch plain round tip. Pipe into seventy two 2-inch-long fingers, spacing them 3 inches apart, on baking sheets lined with nonstick baking mats or parchment. Bake until lightly golden with darker edges, 12 to 15 minutes. Transfer cookies to wire racks with a metal spatula, and let cool completely.
Melt chocolate, stirring, in a bowl set over a pan of simmering water. Spread 1 teaspoon chocolate each onto flat side of half the cookies. Sandwich with remaining cookies, pressing gently.
Cookies can be stored at room temperature in an airtight container up to 2 days.