Heat the milk and water in a small saucepan until just warm. Place it in the bowl of your electric mixer and add the melted butter, the yeast and honey, stirring to combine. Let sit for 15 minutes until the yeast is foamy. Add in 2 1/2 cups of the flour, the egg, vanilla, salt, cinnamon and allspice, then turn your mixer onto low speed with paddle attachment. Stir in about 1/2 cup of the grated carrot. Once everything is combined, add in another 1/2 cup of flour and mix. Switch to the dough hook and knead the dough for 6 to 8 minutes, gradually adding in the remaining half cup of flour.
Add the dough to a well oiled bowl (you can totally use bacon grease!!) and cover. Let rise in a warm place for 1 1/2 to 2 hours.
Once the dough has risen, preheat the oven to 350 degrees F. Punch down the down and place it on a floured surface. Roll the dough out into a large rectangle, at least 12 inches long and 6 inches wide. Brush the dough with the melted butter. In a small bowl, combine the brown sugar, sugar, cinnamon, allspice, ginger, nutmeg and salt, then sprinkle it all over top of the dough. Add the finely grated carrot evenly over top. Starting at the bottom, tightly roll the dough up into one long roll. Use a sharp knife to cut the roll into 1-inch pieces. Place the rolls in a baking dish or pan and cover, allowing them to rise for 30 minutes.
After 30 minutes, brush the rolls with melted butter. Bake for 25 to 30 minutes until golden and set on top. Immediately remove from the oven and pour the icing over top. Serve!