Sort and clean the mussels: Under cold running water, scrub shells well, pull off beards toward the hinge, and discard any that are cracked or that stay open when tapped; drain and keep chilled.
Set a large wide pot with a tight-fitting lid over medium heat; add the unsalted butter. When it foams, add the shallots and cook until translucent, 3–4 minutes, then stir in the garlic and cook just until fragrant, 30–60 seconds.
Raise heat to high; pour in the dry white wine, add the fresh thyme and the bay leaf, and bring to a vigorous boil for 1 minute.
Tip the cleaned shellfish into the pot, cover, and steam over high heat 5–7 minutes, shaking the pot once or twice, until most shells have opened and the meats are plump.
Remove from heat, uncover, and discard any shells that stayed closed. Fish out and discard the herb stems and leaf, then stir in the flat-leaf parsley and season with the black pepper.
Ladle the cooked shellfish and broth into warm bowls and serve immediately with lemon and crusty bread.