Put a large saute pan over medium-high heat and add 2 tablespoons oil. Once heated, add the mussels and cover with a lid to quickly saute until they open, about 5 minutes. Remove the mussels from the pan to a large platter or sheet tray to cool slightly, then remove the mussels from the shells to a bowl and set aside. Reserve the cooking liquid in a bowl.
Hollow out the sourdough boule to create a bowl, then cut the innards into large dice. Put the bread bowl into a 12-inch cast-iron pan and arrange the diced bread around it, then drizzle with olive oil. Put into the oven to toast, stirring the diced bread as needed to evenly toast, 5 to 7 minutes.
Add 2 tablespoons oil to the same pan the mussels steamed in and put back over medium-high heat. Add the onions, carrots, garlic, paprika, rosemary and thyme and sweat until soft, about 3 minutes. Add the vinegar and reserved mussel broth and bring to a simmer. Remove from the heat and allow to completely cool before pouring over the mussels. Taste and adjust the seasoning. Fill the toasted bread bowl with the mussels. Garnish with the chives and serve with the toasted cubed bread.