Heat oil in a pan. Add cumin seeds and mustard seeds. When they splutter add onion, curry leaves, green chillies, black peppercorns, carrots and salt and stir.
Cover and cook for two minutes. Add two cups of water and bring to a boil. Cover and cook for two more minutes. Add chopped coriander leaves and continue to cook till the carrots are done.
Strain and reserve the liquid. Cool and grind the carrots with a little water to a fine puree.
Heat the strained liquid in a deep pan, add the puree and stir. Bring the mixture to a boil. Add coconut milk and lemon juice and let it come a boil.
Pour into four individual soup bowls, garnish each bowl with a swirl of one teaspoon of coconut milk and a sprig of coriander leaves and serve piping hot.