Cut the tomatoes in half and place them on a baking pan. Drizzle 2 tablespoons of olive oil on the tomatoes and season with salt and pepper. Roast the tomatoes for 20-25 minutes or until slightly charred and softened.
Heat 2 tablespoons of oil in a medium saucepot over medium heat. Add the chopped onions and scallions and saute until softened, about 5 minutes. Add the lemongrass, ginger, garlic, and saute for another 30 seconds or until aromatic. Add the tomato paste and cook for 2 minutes or until the tomato paste is caramelized.
Add the roasted tomatoes to the pot along with the star anise, cinnamon stick, bay leaf, stock, and Shaoxing wine. Cover the pot and bring it to a boil. Reduce the heat to low and continue to simmer for 20 minutes.
Meanwhile, make the umami braised mushrooms. Heat ¼ cup oil in a medium saute pan over medium-high heat. Add the halved shiitake mushrooms with the cut side facing down. Cook until the mushrooms are browned, about 4-5 minutes. Remove the mushrooms from the pan and set them aside.
In the same pan, add soy sauce, mirin, sake, brown sugar, and black pepper. Bring to a boil, then add the mushrooms back to the pan and toss to coat. Reduce the heat to low and continue to simmer the mushrooms for 7-8 minutes or until softened and the sauce is almost all evaporated. Keep warm.
Return to the soup. Remove lemongrass and bay leaf. Season the stock with soy sauce, fish sauce, salt, and pepper. Add the butter, stirring until it's fully melted and incorporated into the broth.
Bring a large pot of water to a boil. Add the bok choy and cook until tender, about 2 minutes. Remove the bok choy and in the same pot, cook the udon noodles according to the directions on the package. Portion the noodles into 4 bowls and ladle the soup on top. Top with the bok choy and umami braised mushrooms and garnish with chopped basil, scallions and crispy shallots. Serve immediately.
Notes
The broth can be prepared up to 3 days in advance.