Preheat your oven to 300°F. This low cooking temperature is essential for achieving tender beef cheeks that melt in your mouth.
Sear the beef cheeks in olive oil until they turn a rich, golden brown. Aim for about 4-5 minutes per side to develop deep, savory flavors.
Combine soy sauce, hoisin sauce, sweet soy sauce, and water (or broth) in a bowl. This mixture will create a flavor-packed braise that envelopes the beef.
Return the browned beef to the pot and add the lemongrass, ginger, garlic cloves, red chili, and mushrooms. Stir gently to coat everything in the flavorful liquid.
Braise covered for 2½ to 3 hours, checking occasionally. The beef should become fork-tender and infused with aromatic goodness.
Simmer the carrots in vegetable broth with the 1-inch piece of ginger for about 20 minutes until they are soft and easily pierced with a fork.
Blend the carrots with butter and a pinch of salt until smooth to create a delightful purée that pairs beautifully with the beef.
Mix together the ingredients for the chili-lime sauce in a blender until smooth. This vibrant sauce will add an exciting kick to the dish.
Serve the beef over the creamy carrot purée, drizzle with chili-lime sauce, and garnish with fresh cilantro for a stunning presentation.