Preheat the oven to 350°F and lightly grease an 8-inch cast-iron skillet or baking dish.
In a large mixing bowl, stir together the flour, sugar, brown sugar, baking soda, cinnamon, nutmeg, and salt until well combined.
Pour into the bowl the vinegar, vanilla, oil, coffee, and orange juice, then stir until everything is well combined and a dark batter is formed. It will be slightly fizzy, and this is good!
Stir in the carrot and pecans, then pour the batter into the prepared skillet or baking dish.
Cook the cake uncovered in the oven for 40-45 minutes or until browned on top and an inserted knife comes out clean.
Meanwhile for the frosting, stir together the powdered sugar, orange juice, orange zest, and cinnamon. After the cake has rested for 10 minutes but is still warm, spread the frosting over the cake.
Serve the cake immediately or at room temperature. It will also keep covered for 3 days.