Put the shiitakes in a bowl and add the hot water. Cover and let stand until softened, about 20 minutes. Using a slotted spoon, remove the mushrooms from the water and cut off and discard the stems. Swish the mushrooms in the soaking water to loosen any grit. Finely chop the mushrooms; reserve the soaking liquid.
Heat the oil in a saucepan. Add the mushrooms and cook over high heat until starting to brown, about 3 minutes. Add 2 tablespoons of the soy sauce and the garlic and cook over high heat until the soy sauce evaporates and the mushrooms brown, about 5 minutes. Pour in the reserved soaking liquid, stopping when you reach the grit, and the remaining 2 tablespoons of soy sauce and stir well. Cover and simmer over low heat until the mushrooms are tender, about 5 minutes. Season with salt and pepper.
The jus can be refrigerated overnight. Rewarm before serving.