Melt the butter in a 3- to 4-quart saucepan over medium heat. Add the shallots and garlic and cook, stirring frequently, until both are very soft, 6 to 8 minutes. They shouldn’t brown. If they’re cooking too fast, reduce the heat to medium low.Add the peas, broth, half of the mint, and 2 cups water. Season generously with salt and pepper. Bring to a boil, reduce the heat to medium low, and simmer vigorously until the peas are very tender, 8 to 10 minutes. In batches, purée the soup in a blender until smooth. Season to taste with salt and pepper. If the peas weren’t very sweet, stir in the sugar. Pour the heavy cream into a medium bowl and whip it to soft peaks with a whisk. Fold in the lemon zest and season to taste with salt and pepper. Ladle the soup into serving bowls and top with a generous spoonful of the lemon cream. Scatter the remaining chopped mint over the soup and serve. If you choose to serve the soup cold, chill it in the fridge but take it out 15 minutes before you serve. Adjust the seasoning if necessary before serving.