Chubby Hubby - Slow-cooking an egg

Ingredients

• 6 ounces flat-leaf parsley, leaves only
• 1/2 pound white mushrooms, coarsely chopped
• 1 tablespoon vegetable oil
• Salt and freshly ground pepper
• 2 tablespoons water
• 1/2 cup heavy cream
• 2 tablespoons unsalted butter
• 1 small shallot, thinly sliced
• 1/4 pound small chanterelles or other wild mushrooms, sliced 1/4 inch thick
• 4 large eggs