Combine the water, onion, celery, lemons, jalapeno, garlic, thyme, bay leaves, and crab boil in a large, heavy pot over high heat. Season the water with salt and cayenne and bring to a boil. Add the crabs and cook, covered, for 15 minutes. Drain and transfer the crabs to a large bowl or bowls of ice water. Cool for about 5 minutes, then drain. Set aside.
Combine the green onions, chopped garlic, and chicken stock in a large saucepan over medium heat. Season with salt and cayenne. Bring to a gentle boil and simmer for 10 minutes. Remove from the heat. In a blender or using a handheld blender, process the sauce until smooth. Strain through a fine-mesh strainer. Taste for seasoning, adding more salt and cayenne if necessary.
To serve, pile the crabs on a platter and serve with the dipping sauce on the side.