Preheat oven to 350 degrees, with racks in middle and bottom thirds. Whisk together flours, brown sugar, baking soda, 3/4 teaspoon cinnamon, and salt. In a large bowl, with a mixer, beat yogurt, oil, and eggs on medium, 3 minutes. With mixer on low, add flour mixture in 3 additions; beat in vanilla until combined. Refrigerate batter 30 minutes. Meanwhile, using clean mixer, beat cream cheese until smooth. Beat in confectioners' sugar and lime zest and juice until smooth. Refrigerate filling, covered, until ready to use (or up to 3 days).
Combine remaining 1/4 teaspoon cinnamon and granulated sugar. Drop dough in 40 rounded tablespoons, 1 1/2 inches apart, on parchment-lined baking sheets. Bake until puffed and set, about 10 minutes, rotating sheets halfway through. Remove cakes from oven and immediately sprinkle with cinnamon-sugar. Let cool completely on sheets on wire racks. Peel cakes from parchment. Spread 1 tablespoon filling on flat sides of half the cakes and sandwich with remaining cakes. Serve immediately (or refrigerate in a single layer in an airtight container, up to 2 days).