Saute onion in 1 tbsp melted butter in a large saucepan over medium-high heat, stirring constantly, until tender. Add the chicken broth and bring to a boil. Stir in rice; cover, reduce heat, and simmer 15 to 20 minutes or until rice is tender and the liquid has been absorbed. Set aside, and let cool. Stir in parsley.
Work with 1 sheet of phyllo at a time, keeping the remaining sheets covered with a slightly damp towel so they don't dry out. Layer phyllo sheets on a work surface, brushing each sheet with the remaining 3/4 cup plus 3 tablespoons melted butter. Spoon the reserved rice mixture over phyllo, spreading to within 1 inch of sides. Sprinkle the ham over the rice mixture, then top with the cheeses.
Starting at the short side, fold the phyllo over 1 inch; fold each long side over 1 inch. Roll up phyllo, starting with the short side. Place the phyllo, seam side down, on a lightly greased broiling rack in a broiler pan. Brush with any remaining melted butter. Make several diagonal slits, about 1/4 inch deep, across the top of the pastry with a sharp knife. Sprinkle with paprika. Bake uncovered at 375 degrees for 35 to 40 minutes or until golden.
Let stand 5 minutes before serving.