1. Prepare the Anchovy Vinaigrette:

In a small bowl, whisk together the minced anchovies, Dijon mustard, minced garlic, lemon zest, red wine vinegar, olive oil, salt, and pepper until well combined. Set aside.

2. Season and Sear the Lamb Shanks:

Preheat your oven to 325°F (160°C).

Pat the American lamb shanks dry and season them generously with salt and pepper.

Heat olive oil in a large, ovenproof pot or Dutch oven over medium-high heat. Sear the lamb shanks on all sides until browned, about 3-4 minutes per side. 

Remove the shanks and set them aside.

3. Build the Braising Liquid:

In the same pot, add the chopped onion and garlic, sautéing until softened and translucent, about 5 minutes.

Pour in the white wine, scraping up any browned bits from the bottom of the pot. Let the wine simmer for 2-3 minutes.

Add the broth, rosemary sprigs, and bay leaves. Return the lamb shanks to the pot, ensuring they are mostly submerged in the liquid. Brush the tops of the lamb shanks with some of the anchovy vinaigrette.

4. Braise the Lamb Shanks:

Cover the pot with a lid and transfer it to the preheated oven. Braise the lamb shanks for about 2.5 hours, or until they are tender and nearly falling off the bone.

5. Add the White Beans:

When the lamb shanks have about 30 minutes of cooking time left, add the drained and rinsed white beans to the pot. Stir in the chopped rosemary and the rest of the anchovy vinaigrette.

Continue braising until the beans are heated through and have absorbed some of the braising liquid and flavors, and the lamb is fully tender.

6. Serve:

Remove the pot from the oven. Discard the rosemary sprigs and bay leaves.

Taste the beans and adjust the seasoning with additional salt, pepper, or more anchovy vinaigrette as needed.

Serve the lamb shanks on a bed of the rosemary white beans, with lemon wedges and fresh parsley for garnish.