Preheat oven to 350 degrees. Grease a bundt pan with butter or oil.
Make the streusel. In a small bowl, toss together the sugar, flour, and cinnamon. Cut in the butter with your fingers, two forks, or a pastry cutter until pieces no larger than the size of a pea remain. Set aside.
In a large bowl, beat together the butter and brown sugar until light and fluffy, about 2 minutes. Add eggs, one at a time, until fully incorporated. Beat in the vanilla and yogurt, then the shredded carrots until mixed thoroughly.
Add the flour, baking powder, baking soda, cinnamon, ginger, clove, and salt. Whip until just combined. Slowly stream in the apple juice and beat until a thick batter forms. Do not overmix.
Pour half of the batter into the prepared pan and spread gently with a spatula. Sprinkle on half of the streusel mix. Top with the remaining batter (don't worry if it isn't perfect!) and finish with the rest of the streusel. Place pan in the oven and bake for 55-60 minutes or until a toothpick emerges clean.
Let the cake cool for 30 minutes in the pan. Gently run a knife around the edges and invert the cake onto a plate. Flip the cake upright and cool completely.
When the cake is cooled, prepare the frosting. Melt the butter in a saucepan until golden and nutty, about 5 minutes. Let cool for 10 minutes. Beat the browned butter with the cream cheese and vanilla on low speed until smooth. Add the powdered sugar 1/2 cup at a time and beat until a drizzly but thick icing forms. If needed, add 1-2 tablespoons of milk or water to thin.
Drizzle the glaze over the cooled coffee cake. Slice and serve!