Bring a medium saucepan of heavily salted water to a boil over medium-high heat. Rinse rice and add to boiling water. Cook until rice is slightly underdone, drain, rinse under cold water until cool and set aside.
Remove chard leaves from the stem. Wash stems and leaves thoroughly. Chop each and set aside, keeping them separated.
Heat butter and oil in a large stock pot over medium-high heat. Once butter is melted add leeks, celery, and chard stems. Season with salt and pepper and cook, stirring occasionally, until softened. Add garlic and cook another minute. Add flour and cook, stirring constantly until the flour smells like cooked pie crust. Slowly add chicken stock, stirring continuously, then add chard leaves.
Bring to a boil, then reduce heat to a simmer. Cover and cook until vegetables are tender, about 45 minutes. Puree with a stick blender or in batches in a blender. Stir in milk and cooked rice and return to a simmer. Taste and add more salt and pepper as needed. Serve.