Heat 2 tablespoons of oil in a large lidded saucepan.
Add the shallot, thyme, and mussels and cook for 1 minute.
Pour in the wine, cover, and cook for 3 minutes until mussels open.
Strain mussels through a sieve, reserving both the mussels and the cooking liquid.
Extract half the mussels from their shells and set the rest aside for serving.
Heat remaining oil and butter in a clean saucepan and sauté squash for 2 minutes.
Add stock, shelled mussels, and reserved liquid, then simmer for 8 minutes.
Blend the mixture until smooth and return to the pan to season with salt and pepper.
Fry sourdough cubes in oil until golden-brown and drain on kitchen paper.
Ladle soup into bowls and garnish with croutons, whole mussels, cress, and oil.