Preheat oven to 375 degrees F. Use kitchen shears or a long, heavy knife to cut hen in half lengthwise. Cut through the breast bone, just off center. Cut on each side of the backbone to completely remove the backbone; discard backbone. Remove and discard as much skin as you can from the hen halves.
Place hen halves, cut sides down, in a 2-quart rectangular baking dish. Brush tops with oil. In a small bowl combine salt, paprika, and cumin; sprinkle evenly over hen halves. Bake, uncovered, for 40 to 50 minutes or until hens are tender and no longer pink (170 degrees F in breast, 180 degrees F in thigh).
Meanwhile, coat a small saucepan with nonstick spray. Heat over medium-high heat. Add mushrooms, celery, green onion, and sage. Cook for 5 to 8 minutes or until mushrooms and celery are tender, stirring frequently.
Squeeze pouch of rice to break up rice; open pouch. Measure 1 cup of rice and add to mushroom mixture (reserve remaining rice for another use). Cook and stir rice mixture about 2 minutes or until heated through. Serve with hen halves. If desired, garnish with fresh thyme sprigs.