Heat oil in a 12-inch nonstick skillet over medium-high heat to 3750 F. Carefully add tempeh to hot oil and increase heat to high. Cook, turning tempeh as needed, until golden brown, 3 to 5 minutes, adjusting burner if necessary to maintain oil temperature between 350°F and 375°F degrees. Transfer tempeh to paper towel-lined plate.
Add the shimeji to the oil and stir fry until golden brown. Add them to the plate with the tempeh.
Carefully pour off all but 2 tablespoons oil from pan. Add sambal oelek to the remaining oil in the skillet and cook over medium-high heat, stirring slowly but constantly, until darkened in color and a dry paste forms, 7 to 10 minutes. Remove from the heat heat and stir in water and kecap manis until well combined. Add tempeh and basil and toss until well coated. Serve with steamed rice.
Serves 2, but is easily doubled.