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Preheat oven to 350F. Line 2 baking sheets with parchment paper.
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Combine shredded carrots and 3 tablespoons brown sugar in a bowl; toss to coat. Place carrot mixture in a fine mesh sieve; let stand for 10 minutes. Discard liquid.
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Microwave 2 tablespoons butter in a medium microwave-safe bowl until melted, about 45 seconds. Add oil, orange zest, ½ teaspoon vanilla, and egg. Whisk until well combined.
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In a separate large bowl, combined flour, remaining brown sugar (about ½ cup), cinnamon, salt, and baking soda; whisk together. Add carrot mixture and butter mixture; stir until everything is incorporated.
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Drop dough by tablespoonfuls 2 inches apart onto prepared baking sheets. You should get 28 cookies. Gently pat dough down to form 2-inch circles.
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Bake for 11 - 13 minutes or until cookies are set. Let cool completely on wire rack.
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Combine cream cheese, remaining 2 tablespoons butter, remaining ¼ teaspoon vanilla, and a pinch of salt in a medium bowl. Beat with a mixer at medium speed until fluffy. Add powdered sugar and milk; beat at a very low speed until combined.
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Spread about 1 tablespoon on flat side of 1 cookie, top with another cookie, flat side down. Repeat with remaining cookies and filling.