Preheat oven to 350°F (175°C).
In a medium saucepan, combine 1 cup dry rice, 2 cups water, 1/2 tsp salt, and 1 tbsp butter. Bring to a boil, then reduce heat to low, cover, and simmer 15–20 minutes until the water is absorbed and the rice is tender. Remove from heat, keep covered for a few minutes, then fluff with a fork and let cool slightly.
While the rice cooks, in a medium mixing bowl beat the 2 eggs with a fork until combined. Add the finely grated medium onion, 1 cup chopped parsley, and 2 cups shredded cheddar; stir to combine.
Add 1 tsp salt, 1/2 tsp dry mustard, 1 1/2 cups milk, and 1/2 cup vegetable or canola oil to the bowl with the egg mixture. Mix until evenly combined.
Add the cooked, slightly cooled rice to the bowl and stir until everything is evenly combined.
Grease a 9 x 11-inch baking dish with a little of the 1/2 cup vegetable or canola oil. Transfer the rice mixture to the dish and spread it evenly.
Bake in the preheated 350°F oven for 1 hour, until set and lightly browned on top. Let rest a few minutes before serving.