Warm 1 cup of milk and ¼ cup of water over low heat until just warm. Combine with 2 tablespoons of melted butter, 3 teaspoons of active dry yeast, ¼ teaspoon of sugar, and 1 tablespoon of honey. Let sit for 15 minutes until bubbly.

In a mixing bowl, whisk together 3 ¼ cups of flour, 1 egg, 1 teaspoon of maple extract, ½ teaspoon of cinnamon, ½ teaspoon of allspice, and 1 ¼ cups of grated carrot. Stir in the yeast mixture, kneading until smooth and elastic, about 5-6 minutes.

Transfer the dough to a greased bowl, cover with plastic wrap, and let rise in a warm area for 1.5 to 2 hours until doubled in size.

Preheat the oven to 350°F. Punch down the dough and roll it out into a large rectangle (about 12x18 inches). Brush with melted butter and sprinkle the filling mixture evenly on top.

Roll the dough tightly from one long side to the other, cut into 1-inch thick slices, and arrange in a greased baking dish. Let them rise for an additional 30 minutes.

Brush the tops with any remaining melted butter and bake for 25-30 minutes until golden and cooked through.

Whisk mascarpone until creamy, gradually sift in powdered sugar, then add maple extract and milk until desired consistency.

Allow rolls to cool slightly before drizzling with icing and serve warm.