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Preheat your oven to 200°C (400°F). Season the lamb shanks evenly with ½ tbsp salt. Heat an oven-safe Dutch oven or heavy pot over high heat, sear the shanks in their own fat until deeply browned on all sides, then transfer them to a plate.
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In the residual lamb fat over medium heat, sauté the smashed garlic, ginger, and scallions until fragrant and lightly golden. Stir in the brown sugar, cinnamon stick, star anise, and bay leaves, letting the sugar melt and caramelize briefly with the aromatics.
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Pour in both soy sauces and broth, stirring to scrape up any browned bits from the bottom. Return the lamb shanks to the pot, cover with a tight-fitting lid, and braise in the oven for 2 to 2.5 hours, or until the meat is fork-tender and falling off the bone.
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Carefully remove the tender lamb shanks and keep them warm on a platter. Place the pot back on the stove over medium-high heat and boil the braising liquid uncovered until reduced by half into a glossy, rich sauce.
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Bring a separate pot of water to a boil. Cook egg noodles according to package directions and blanch the choi sum until tender-crisp. Drain both thoroughly, then toss them with the sesame oil to prevent sticking and add flavor.
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Divide the sesame-dressed noodles and choi sum between two bowls. Top each with a braised lamb shank and spoon a generous amount of the reduced braising sauce over everything.