In a large Dutch oven over medium heat, melt the butter.
Add the leeks, carrots, celery, and garlic. Cook, stirring often, until softened and fragrant, about 5 minutes.
Sprinkle the flour over the vegetables and stir constantly until the raw flour smell disappears, about 1 minute.
Increase the heat to medium-high. Slowly add the chicken broth, then stir in the shredded chicken and thyme. Bring to a boil.
Reduce the heat to medium-low and simmer until the flavors meld and the soup thickens slightly, about 10 minutes.
Add the collard greens and cook until just tender, about 5 minutes.
Remove the pot from the heat. Stir in the cooked rice, heavy cream, salt, and black pepper until well combined.
Taste and adjust seasoning as needed. Serve warm, garnished with extra fresh thyme.