Place the shiitake mushrooms in a bowl and pour in warm water. Soak for 15 minutes, flip them over and soak for 15 minutes more (soaking them for as long as three hours won't hurt).
Snip off the stems of the shiitake and discard. Cut the caps into halves if large. You want bite-size pieces.
Strain the liquid in which the mushrooms have soaked. You will need about three cups.
While the shiitake mushrooms rehydrate, rinse the sea cucumbers and make sure the cavities are clean.
Drop the sea cucumbers into a pot, add half of the ginger and cover with water. Bring to the boil, lower the heat, cover the pot and cook the sea cucumbers with the ginger for 15 to 20 minutes to remove the strong fishy odor.
Drain the sea cucumbers, rinse under the tap (or plunge in an ice bath) then cut into bite-size pieces.
Heat the cooking oil in a wok or frying pan.
Saute the remaining ginger, all of the garlic and shallot(s) in the oil until fragrant.
Turn up the heat. Add the shiitake and sea cucumbers. Stir fry until every piece is coated with oil.
Reserve a quarter cup of the mushroom liquid and pour the rest into the wok with the sea cucumbers and mushrooms.
Stir in the soy sauce, pepper, sugar and oyster sauce. Bring to the boil. Set the heat to low, cover the wok and cook the sea cucumbers and mushrooms until tender and the liquid is reduced to less than half.
Stir the starch in the reserved mushroom liquid.
Pour the starch solution into the wok. Cook, stirring, until the sauce is thick and clear. Taste. Add salt, as needed.
Off the heat, drizzle in the sesame seed oil and toss everything to blend.
Arrange the blanched broccoli florets on one side of a serving platter. Ladle your delicious Chinese-style braised sea cucumbers and shiitake mushrooms, with the sauce, on the opposite side.
Serve at once.