Heat oven to 250 F with a rack in the middle position.
Twist a three-foot-long piece of foil into a snake and coil it so that it fits inside a 9x13 inch baking dish. Place the bamboo sushi mat on top of the coil and fold the sides in to make it fit snugly.
Line a fine mesh strainer with a few layers of cheesecloth and position over a medium bowl or large measuring cup. Gently whisk the eggs, soy sauce, mirin, and salt together in another bowl, then slowly add the dashi, whisking to combine. Pour the mixture through the lined strainer, discarding any bits that may be left behind.
Bring 3 cups of water to a boil in a small pot, or electric kettle.
Evenly distribute the custard mixture among the 4 bowls, wrap each individually with plastic wrap, and place atop the sushi mat. Position the baking pan on the middle rack of the oven and carefully pour the boiling water into one corner of the baking dish. (It should come up a half inch and not go above the bamboo mat.)
Place the plastic-wrapped bowls on the sushi mat, then cover the whole pan tightly with foil. (You may need 2 pieces to properly cover it.) Bake for 1 hour.
While the chawanmushi is steaming, cook the shimeji mushrooms: Place the butter in a small skillet over medium heat and cook until foaming. Add the mushrooms and sauté, constantly tossing, for 1 minute until lightly browned. Kill the heat and add the remaining 1/2 teaspoon of soy sauce, tossing to coat. Remove the mushrooms to a small paper towel lined plate.
When the hour is up, carefully remove the chawanmushi from the oven and remove the foil. Lift a small edge of plastic wrap from one of the bowls (some steam will escape, so use caution) and insert a digital thermometer into the center of the custard. Temp should be around 175 F (174 F-177 F range). Let it rest for 5 minutes before removing the plastic wrap from the rest of the bowls.
Garnish with the shimeji mushrooms and other toppings of your choice and serve. Chawanmushi can be eaten warm, at room temperature, or chilled. (I prefer mine slightly warmer than room temp.)