Heat a large pot over medium heat. Once the pot is hot, add the olive oil. Trim the base off the bok choy and cut the bok choy in half. Sear both sides of each half, about 2 minutes, until slightly brown and tender. Remove and set aside.
In the same pot, add the onions, garlic and ginger. Cook until the edges of the onions start to brown, about 2 minutes. Add the brown beech mushrooms and cook until they begin to soften, about 1 minute. Add ¼ cup of the broth to deglaze the brown bits on the bottom of the pot. Add the rest of the broth and simmer for about 20 minutes (though you can cook longer for a deeper flavor).
Add the sliced carrots, and cook until tender, about 10 minutes.
Meanwhile in a separate pot, fill with water and bring to a rolling boil. Add the noodles and gently separate with chopsticks or a fork. Cook for 2 ½ minutes. Strain the noodles, then run them under cold water briefly to prevent overcooking.
Add the noodles and bok choy to serving bowls and pour the broth over the top. Garnish with radishes, cilantro, green onions and a squeeze of lemon over each bowl. Add salt and pepper to taste.
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