Rinse the mussels under cold running water, scrubbing away any debris and removing the 'beards' if present. Discard any mussels that are cracked or remain open when tapped.
Mince the garlic and finely dice the shallot.
In a large pot or deep skillet, melt the ghee over medium heat. Add the shallots and sauté for 2-3 minutes until translucent, then add the garlic and cook for another 30 seconds until fragrant.
Pour in the white wine and chicken broth, bringing the liquid to a gentle simmer for 2 minutes to allow the flavors to meld.
Add the mussels to the pot and immediately cover with a tight-fitting lid. Steam for 5-7 minutes, or until all the mussels have opened wide.
While the mussels steam, toast the sourdough bread slices until golden and crisp.
Remove the pot from the heat and discard any mussels that did not open. Stir in the fresh chopped parsley, sea salt, and black pepper.
Squeeze the juice of the half lemon over the mussels and stir gently to incorporate.
Transfer the mussels and the garlic-butter broth to a large bowl and serve immediately with the toasted sourdough bread on the side.