Bring water, rice, and pinch of salt to a boil in a saucepan. Reduce heat to medium-low; cover, and simmer until kernels burst open, 45 to 50 minutes, adding more water if necessary. Fluff rice with a fork; let stand 5 to 10 minutes to absorb any excess liquid.

Preheat the oven to 425 degrees F (220 degrees C). Grease a baking dish large enough to hold hen.

Melt 2 tablespoons butter in a skillet over medium heat. Add mushrooms; cook and stir until partially cooked and starting to release liquid, 5 to 8 minutes. Stir mushrooms into rice.

Season hen, inside and out, with poultry seasoning, rosemary, salt, and pepper. Lightly stuff hen with wild rice mixture; tie legs together with cooking twine. Place hen in prepared baking dish; add 2 tablespoons butter pats around hen.

Bake in the preheated oven until no longer pink at the bone and the juices run clear, 40 to 50 minutes. An instant-read thermometer inserted into the thickest part of the thigh, near the bone but not touching should read 165 degrees F (74 degrees C). Transfer to a cutting board, cover with a doubled sheet of aluminum foil; rest in a warm area 10 minutes before slicing.