In the bowl of a stand mixer (or a large mixing bowl), combine the warm milk, yeast, and granulated sugar. Let it sit for 5-10 minutes until foamy, indicating the yeast is active.

Add the melted butter, egg, and salt to the yeast mixture. Mix well. Gradually add 3 cups of flour, mixing on low speed until combined. Add the remaining flour ½ cup at a time until a soft, slightly sticky dough forms. You may not need all 4 ½ cups.

Knead the dough with the dough hook on medium-low speed for 5-7 minutes, or by hand on a lightly floured surface for 8-10 minutes, until the dough is smooth and elastic.

Lightly grease a large bowl with oil. Place the dough in the bowl, turning once to coat. Cover the bowl with plastic wrap or a clean kitchen towel and let it rise in a warm place for 1 to 1 ½ hours, or until doubled in size.

While the dough is rising, prepare the carrot filling: In a medium bowl, combine the grated carrots, brown sugar, melted butter, cinnamon, ginger, and nutmeg. Mix well and set aside.

Once the dough has doubled, punch it down gently to release the air. Turn the dough out onto a lightly floured surface and roll it into a large rectangle, approximately 12x18 inches (30x45 cm).

Spread the carrot filling evenly over the entire surface of the dough, leaving a small ½-inch border on one of the long edges.

Starting from the long edge opposite the border, tightly roll up the dough into a log. Pinch the seam closed to seal.

Using a sharp knife or dental floss, cut the log into 9 equal slices (approximately 1 ½ to 2 inches thick).

Lightly grease a 9x13 inch (23x33 cm) baking dish. Arrange the sliced rolls in the prepared dish, leaving a little space between each.

Cover the dish loosely with plastic wrap or a kitchen towel and let the rolls rise in a warm place for another 30-45 minutes, or until puffy and almost doubled in size.

Preheat your oven to 375°F (190°C).

Bake the rolls for 20-25 minutes, or until golden brown and cooked through. If they start to brown too quickly, you can loosely tent them with foil.

While the rolls are baking, prepare the rum-butter syrup: In a small saucepan, combine the butter, brown sugar, and water. Bring to a simmer over medium heat, stirring until the sugar is dissolved and the butter is melted. Remove from heat and stir in the dark rum and vanilla extract.

Immediately after removing the rolls from the oven, pour the warm rum-butter syrup evenly over all the hot rolls. Allow them to sit and absorb the syrup for 10-15 minutes.

While the rolls are absorbing the syrup, prepare the cream cheese glaze: In a medium bowl, beat the softened cream cheese until smooth. Gradually add the sifted powdered sugar, mixing until creamy. Stir in 1 tablespoon of milk/cream and vanilla extract. Add more milk/cream, ½ teaspoon at a time, until you reach your desired drizzling consistency.

Drizzle the cream cheese glaze generously over the warm, syrup-soaked rolls.

Serve warm and enjoy your Drunk Bunny Carrot Rolls!