In a small bowl, combine dried mushrooms and 1 cup boiling water. Let stand 20 minutes until softened. Strain the mushrooms, saving the soaking liquid. Finely chop mushrooms.

In a food processor, process bread until finely crumbed; set aside. Using the food processor again, coarsely chop the fresh mushrooms; set aside.

In a large saucepan, heat 1 tablespoon of the olive oil over medium heat. Add onion, carrot and garlic and cook, stirring frequently, until onion is tender, about 5 minutes. Add fresh chopped mushrooms and reconstituted dried mushrooms along with salt, thyme, black pepper and rice. Turn heat to medium-high and stir until the mushrooms release their liquid, about 3 minutes. Add mushroom soaking liquid. Bring to a boil. Cover and reduce heat to low. Cook until rice is tender and liquid has been absorbed, 50 to 55 minutes. Stir in bread crumbs.

Transfer mixture to a bowl. Let cool to room temperature or until it is cool enough to handle.

In a large non-stick skillet, heat remaining tablespoon of olive oil over medium heat. Shape mushroom mixture into six patties. Cook 3 to 5 minutes per side over medium heat until crusty and heated through. Cook in batches if needed so as not to crowd the burgers. Just before serving, top burgers with cheddar cheese.