In a large saucepan, bring 2 3/4 cups water, the rice and 1/4 tsp. salt to a boil over high heat. Lower the heat, cover and simmer until the rice is tender, about 20 minutes. Spread the rice out on a baking sheet to cool.
Meanwhile, in a large skillet, melt 3 tbsp. butter over medium heat. Add the onion and cook, stirring, until softened, about 3 minutes. Add half of the mushrooms and cook until the liquid has evaporated and the mushrooms are browned, about 10 minutes. Transfer the mushrooms to the baking sheet with the rice.
In the same skillet, heat the remaining 2 tbsp. butter and the oil. Add the remaining mushrooms and the ham and cook, stirring frequently, until the liquid has evaporated and the mushrooms and ham are browned, about 8 minutes.
Return the rice and mushroom mixture to the skillet and cook, stirring frequently, until heated through, about 3 minutes. Stir in the tarragon and season with salt and pepper.