Stir together all the ingredients until fully incorporated. Use immediately or refrigerate in a sealed container for up to 1 week.
Place the tempeh in a freezer bag, or a plastic bag, and cover with the marinade. Squeeze any remaining air out of the bag and allow the tempeh to marinate for at least 2 hours. To cook, heat a grill or a cast-iron pan over high heat. Remove tempeh from the marinade and sear on both sides, being careful not to burn it, about 4 minutes per side. Cut tempeh into ½-inch cubes or crumble.
Heat a sauté pan over medium-high heat and place the mushrooms in it. After about a minute, add oil and toss the mushrooms in pan. Cook for another 1 minute, then add the tamari and vinegar, and toss together. Remove from heat and set aside.
Combine all ingredients in a high-speed blender and blend until smooth, about 2 minutes.
Heat the kombu, water, and dried shiitakes in a large pot. Bring the water almost to a simmer, then turn it down to low. The water should have steam coming off the top, but not reach a simmer. Cook for 40 minutes, stirring occasionally. Pull from the heat and place in warm place. Cover the pot and let the broth steep for an hour. Whisk in chickpea miso. Strain through a fine mesh strainer.