Fill both the top and bottom halves of the clay pot with cold water and let them stand while you prepare the vegetables.
Cover the dried mushrooms with the boiling water and set them aside to soak. Trim the artichokes: Break off the tough outer leaves and slice off the remaining cone of inner leaves. Trim the outside and scoop out the choke with a spoon. As you work, put the cut artichokes in a bowl with lemon juice and water to cover to keep them from browning. Slice the trimmed hearts into pieces about the same width as the potatoes.
Remove the mushrooms from the soaking water and run your fingers over the surfaces to loosen any grit; then squeeze them dry. Cut them into bite-sized pieces. Let the soaking water stand until the particles settle to the bottom.
Empty the clay pots then add the potatoes, artichokes, garlic cloves, and mushrooms. Season with salt and pepper and toss with the oil, if using, and the herbs. Carefully pour in the soaking water from the mushrooms.
Cover the pot with the lid and