Put the mushrooms in a bowl that holds at least 3 cups. Pour the boiling water over the mushrooms. Let sit.
Cut the corn off the cobs. Put the cobs in a pot. Put the kernels in another pot.
Add enough water to the pot with the cobs to cover them. Add ½ tsp salt, the bay leaves, and rosemary.
Heat the cobs on high to bring to a boil. Reduce the heat to keep at a simmer.
Peel and grate the carrots. Cut them into ¼ inch thick rounds. Add the carrots to the pot with the corn kernels.
Finely dice the radishes. Add them to the corn and carrots.
Peel the potato. Cut it into ½ inch cubes. Add the cubes to the pot with the veggies.
Add the oil, remaining salt, black pepper, and cayenne pepper to the veggie pot. Stir well.
Cook the veggies with the heat just above medium low for 30 – 35 minutes.
Transfer half of the veggies to a blender. Add 1 cup of water from the mushroom bowl and 2 cups of water from the corn cob pot. Blend until mostly smooth.
Pour the blended veggies back into the pot with the rest of the veggies.
Drain the water from the mushrooms. Add the mushrooms to the veggie pot.