Step 1: Heat the olive oil in a heavy-bottomed pot or Dutch oven over medium-high heat until shimmering.
Step 2: Carefully place the lamb shanks into the hot oil. Sear them on all sides until beautifully browned and caramelized, about 3-5 minutes per side. Remove shanks and set aside.
Step 3: Reduce heat to medium. Add minced garlic to the pot and stir for about 1 minute until aromatic. Do not burn.
Step 4: Pour in the red wine. Use a wooden spoon to scrape the bottom of the pot, loosening any browned bits. Bring to a gentle simmer and let bubble for 2-3 minutes to reduce slightly.
Step 5: Carefully place the browned lamb shanks back into the pot.
Step 6: Pour in the beef broth, ensuring the shanks are mostly submerged. Bring the pot back to a gentle simmer.
Step 7: Reduce heat to low, cover tightly, and cook gently for 2 to 2.5 hours, or until the meat is fork-tender and falling off the bone. Alternatively, transfer the covered pot to a preheated oven at 325°F (160°C) for the same amount of time.
Step 8: Once tender, carefully remove the shanks from the pot.
Step 9: Serve each lamb shank over creamy mashed potatoes, spooning the rich pan sauce over the top.