Peel the carrots and ginger, shallots and garlic and cut everything into thin slices. Heat the oil in a pot. Sauté shallots and garlic, add carrots and ginger and sauté.
Sprinkle with sugar and caramelize slightly, stir in the curry powder. Top up with the broth and cover and simmer over a medium heat for about 25 minutes.
Wash the lime with hot water and rub dry. Rub the skin, cut the fruit in half and squeeze the halves. Puree the soup with the hand blender. Season to taste with lime juice and salt.
Heat the coconut milk with the cream in a saucepan and mix until frothy with the hand blender.
Pour the soup into soup plates or cups. Spread the coconut foam on top and sprinkle with the grated lime zest.