Preheat the oven to 350° F. Butter a (9×5 inch) bread pan - line with parchment paper.In a large bowl, stir together the butter, brown sugar, yogurt, eggs, and vanilla until combined. Stir in the grated carrots. Add the flour, baking soda, cinnamon, and salt. Mix until combined.To make the cinnamon swirl. Mix the butter, brown sugar, flour, and cinnamon.Spoon half the batter into the prepared bread pan. Sprinkle the cinnamon swirl over the top. Evenly spoon the remaining batter over the cinnamon. Bake for 55-60 minutes or until the center is just set. Remove the bread and let cool.To make the icing. In a medium bowl, beat together the powdered sugar, cream cheese, lemon juice, and 1-3 tablespoons of water until smooth and drizzly. Spoon over the cake. Mix the sugar and lemon zest. Sprinkle the lemon sugar over the cake. Let the icing set for 30 minutes. Slice and enjoy!