PREHEAT the oven to 350ºF. Butter and flour a 9x5-inch loaf pan and set aside.
IN a large mixing bowl beat the butter, yogurt, and brown sugar together until light and fluffy. Then, beat in the eggs and vanilla extract. Stir in the carrots.
ADD the flour, baking soda, baking powder, salt, and cinnamon to the wet ingredients and stir everything together so that it is smooth with no lumps. Then, pour the batter into the prepared loaf pan and level the top with a spatula.
BAKE for 40 to 45 minutes, or until the top springs back when touched and a cake tester comes out clean. Allow the loaf to cool for ten minutes in the pan. Then, turn it out onto a wire rack and allow it to cool completely.
ONCE it has cooled beat together the cream cheese, sugar, and milk until a thick glaze forms. If it needs to be thinner add more milk by the teaspoon. Drizzle the glaze over the loaf as desired. Slice and serve.