Combine first 3 ingredients in a bowl; cover and let stand 30 minutes or until soft. Drain in a colander over a bowl, reserving 1 cup soaking liquid. Chop tomatoes; set aside chanterelles and tomatoes.
Steam asparagus, covered, 3 minutes or until crisp-tender; set aside.
Bring the reserved soaking liquid and broth to a simmer in a medium saucepan (do not boil). Keep warm over low heat.
Heat oil in a large nonstick skillet over medium heat. Add shallots, garlic, and cremini mushrooms; sauté 6 minutes or until tender. Stir in rice; cook 1 minute. Stir in chanterelles, tomatoes, wine, salt, and pepper; cook 2 minutes or until wine is nearly absorbed. Stir in 1/2 cup broth mixture; cook 3 minutes or until the liquid is nearly absorbed, stirring constantly. Add remaining broth mixture, 1/2 cup at a time, stirring constantly until each portion of broth mixture is absorbed before adding the next (about 25 minutes total). Add asparagus and cheese, and cook 1 minute. Remove from heat; sprinkle with parsley. Serve immediately.