Place the chicken breasts into a large saucepan. Pour in the water, and season with chili powder, 1/2 teaspoon salt, cumin, 1/4 teaspoon black pepper, garlic powder, onion powder, and cayenne pepper. Bring to a simmer over high heat, then reduce heat to medium-low, and simmer 15 minutes. After 15 minutes, stir in 4 ounces of green chiles, onion, and garlic; continue simmering until the liquid has reduced to 1 cup. Reserve remaining 3 ounces of chopped green chilies. Remove the chicken, shred with two forks, and return to the onion mixture.
Meanwhile, melt the butter in a small saucepan over medium heat. Stir in the flour, and cook for 1 minute. Whisk in the water and bouillon cube until the water has thickened and the bouillon cube has dissolved, about 4 minutes. Whisk in reserved 3 ounces of green chilies and the sour cream; season to taste with salt and pepper. Keep warm.
Heat oil in a deep-fryer or large saucepan to 375 degrees F (190 degrees C).
Place a tortilla onto your work surface, then spoon about 1/3 cup of the chicken filling halfway between the bottom edge and the center of the tortilla. Flatten the filling into rectangle shape with the back of a spoon. Sprinkle about 1 ounce of the Monterey Jack cheese over the filling. Fold the bottom of the tortilla snugly over the filling, then fold in the left and right edges.Roll the chimichanga up to the top edge, forming a tight cylinder; secure the ends with wooden toothpicks. Repeat with the remaining ingredients.
Cook the chimichangas, 2 at a time, in the hot oil until they are crisp and golden brown on both sides, about 1 minute per side. Drain on a paper towel-lined plate, and remove toothpicks. Top with sour cream sauce to serve.
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