Cut bok choy into 1/2 inch pieces, stems and greens separated. Drain and pat dry tofu, and cut into 1/2 inch cubes. Grate carrot.

Cut a 1 inch piece of ginger and then slice it into 4-5 thinner slices; add to Dutch oven with crushed garlic and broth. Bring to a low boil. Simmer partially covered, over medium-low heat for 15 minutes. Discard the ginger and garlic. (This will add a lot of flavor but cut back on the stronger flavors of the garlic and ginger).

Meanwhile, heat oil in a large nonstick skillet over medium-high heat. Add sliced mushrooms and crushed red pepper; cook, stirring often, until tender, 3 to 5 minutes. Add bok choy stems. Cook, stirring often until tender, 3 to 4 minutes.

Add the mushroom mixture to the broth. Add Chinese wheat noodles (or rice sticks), reduce heat to medium-low and simmer for 3 minutes. Add bok choy greens and tofu; simmer until heated through, about 2 minutes. Stir in carrots, vinegar, soy sauce and sesame oil. Serve garnished with thinly sliced green onions and pineapple on the side.