Slice the carrots into thin slices at an angle. Remove the stringy part of the peas. Remove the stem of the mushrooms and scrape out the gills. Cut heads into 2-inch wedges. Remove the stems from the bok choy leaves. Cut the stems into quarter inch strips. Cut the leaves into 2-inch pieces.
Mix the ingredients of the glaze. Mix the ingredients of the sauce.
Heat 2 tbsp oil until simmering in a skillet. Add the mushrooms and cook without stirring for 3 minutes. Turn mushrooms and reduce heat to medium. Cook another 5 minutes until both sides are browned. Add glaze and cook until it coats the mushrooms well (about 2 minutes). Remove mushrooms onto a plate.
Add 1 tbsp of oil to the skillet. Add carrots and cook for 2 minutes, until beginning to brown. Add 0.5 cup of broth and simmer covered for 2-3 minutes, until tender. Remove from skillet.
Add the remaining oil to the skillet. Heat again. Add snow peas and bok choy stems. Cook for 2 minutes. Add the leaves of the bok choy. Add the minced ginger and garlic in the middle of the pan. Cook for 15 seconds, then stir into greens.
Add back all vegetables and pour in the sauce. Cook until thickened. Serve over rice.