In a large bowl, whisk together warm milk, yeast, and 1 tablespoon of the sugar. Let sit until foamy, about 5 minutes. Add remaining sugar, melted butter, egg, flour, salt, and cinnamon. Mix until a shaggy dough forms, then knead by hand or with a dough hook on medium-low speed for 6-8 minutes until smooth and elastic. The dough should feel tacky but not sticky; add flour 1 tablespoon at a time if needed.

Transfer dough to a lightly oiled bowl, turning to coat. Cover with plastic wrap and let rise in a warm spot until doubled in size, about 60-75 minutes. The dough is ready when a gentle poke leaves an indentation that springs back slowly.

While dough rises, grate carrots on the fine side of a box grater. Wrap in a clean kitchen towel and squeeze firmly to remove excess moisture; you want about 3/4 cup after squeezing. Set aside. Mix brown sugar, cinnamon, and ginger in a small bowl.

Punch down dough and turn onto a lightly floured surface. Roll into a 12x16 inch rectangle with the long side facing you. Spread soft butter evenly to the edges, leaving a 1-inch border at the top. Sprinkle brown sugar mixture over butter, then scatter squeezed carrots evenly. Starting from the bottom, roll tightly into a log, pinching the seam to seal.

Slice into 12 even pieces using unflavored dental floss or a sharp serrated knife. Arrange in a greased 9x13 inch dish, spacing slightly apart. Cover loosely and let rise until puffy and touching, about 30-40 minutes. Preheat oven to 350 degrees F.

Bake until golden brown and the internal temperature reaches 190 degrees F, 22-26 minutes. The tops should spring back when lightly pressed. Cool in the pan on a wire rack for 10 minutes.

Beat cream cheese and butter until smooth. Add powdered sugar and vanilla; beat until fluffy, adding milk 1 teaspoon at a time if needed for spreading consistency. Spread generously over warm rolls so it melts slightly into the crevices. Serve warm or at room temperature.