Preheat oven to 350°F. Cover 2 baking sheets with parchment paper; set aside.
Combine carrot and 3 tablespoons brown sugar in a bowl; toss to coat. Place carrot mixture in a fine mesh sieve; let stand to drain for 10 minutes. Discard liquid.
Place 2 tablespoons butter in a medium microwave-safe bowl. Microwave at HIGH 45 seconds or until melted. Add oil, rind, 1/2 teaspoon vanilla extract, and egg; stir with a whisk until well combined.
Weigh or lightly spoon flour into a dry measuring cup; level with a knife. Combine flour, remaining brown sugar (about 1/2 cup), cinnamon, 1/4 teaspoon salt, and baking soda in a large bowl; stir well with a whisk. Add carrot mixture and butter mixture; stir until just combined.
Drop dough by tablespoonfuls 2 inches apart onto prepared baking sheets for a total of 28 cookies (14 per baking sheet); gently pat dough down to form 2-inch circles. Bake at 350°F for 11 minutes or until set. Remove pans from oven; let stand 3 minutes. Remove cookies from pans; cool completely on a wire rack.
Combine cream cheese, remaining 2 tablespoons butter, remaining 1/4 teaspoon vanilla extract, and remaining 1/8 teaspoon salt in a medium bowl; beat with a mixer at medium speed 3 minutes or until fluffy. Add powdered sugar; beat at low speed 1 minute or until well combined (do not overbeat).
Spread about 1 tablespoon icing on flat side of 1 cookie; top with another cookie, flat side down. Repeat the procedure with remaining cookies and filling.