In a small saucepan, bring 1 cup water, the rice and 1/2 tsp. salt to a boil over high heat. Lower the heat to low, cover and cook until the water is absorbed, about 15 minutes. Fluff with a fork and transfer to a medium bowl to cool slightly. Add the beans, 1 cup Monterey jack, the scallions and cream cheese; stir gently.
Meanwhile, using a blender or food processor, blend the salsa verde, cream and 1/2 cup cilantro until smooth.
Preheat the oven to 350 degrees . In a medium skillet, heat the oil over medium-high heat. Using tongs, carefully place 1 tortilla at a time into the hot oil and fry, turning once, until softened, 5 to 10 seconds per side. Transfer to paper-towel-lined plates to drain.
Mound 1/2 cup of the filling down the center of each tortilla. Roll up tightly, then place seam side down in a 7-by-11-inch glass baking dish. Pour the sauce over the top and sprinkle with the remaining 1/3 cup Monterey jack. Cover with foil and bake until bubbly, about 20 minutes. Top with more cilantro.