Heat olive oil and 25g butter in a large lidded saucepan.
Add onion, cover, and fry for 4 minutes.
Add carrots and ground coriander and fry for 3 minutes.
Pour in the stock, bring to a boil, and simmer for 5 minutes until carrots are tender.
Add chopped coriander and blend the soup until smooth.
Heat frying oil to 180°C in a deep fryer or heavy pan.
Boil 50g butter and 125ml water in a saucepan for 1 minute.
Beat in flour and cayenne pepper until smooth and pulling away from the pan.
Cook for 2 minutes while stirring continuously.
Remove from heat and beat in eggs one at a time.
Fold in the white and brown crabmeat and season.
Pipe 1.5cm sections into the hot oil and fry until golden.
Drain beignets on paper and season with salt.
Serve soup in bowls topped with beignets, olive oil, and baby coriander.