In a large stockpot, heat olive oil over medium-high heat. Add shallots and garlic and cook 2 minutes, stirring frequently. Then add diced chorizo, cook an additional minute while stirring. Add wine and simmer till almost evaporated. Stir in the pesto, add the stock, and bring to a simmer.
When broth is simmering, add salt and pepper to taste. Make sure all mussels are tightly closed and not cracked. Discard any mussels that are damaged or won’t close. Place mussels in broth, cover the pot with a lid and cook for approximately 5-7 minutes or until all the shells are open. If any mussels do not open in the cooking process, discard.
Garnish with parsley and use crusty bread to dip into broth and soak up all the delicious flavor.