In a medium pot over medium-high heat, combine 1 ¼ cups vegetable stock with frozen brown rice and kabocha squash. Heat until the rice and squash have thawed and the mixture is simmering.

Carefully pour the mixture into the food processor or blender and blend until smooth. {Be careful when blending hot mixtures. You may want to open the feed a tiny bit to let some steam escape and cover with a kitchen towel while you blend.} While continuing to blend, slowly pour in the milk and continue to blend until fully combined. Set aside.

Heat the oil in the pot over medium heat, then add chopped onion, cooking until softened and translucent, about 4-5 minutes. Add garlic and cook for about a minute.

Add broccoli, salt and pepper, and cook for 2-3 minutes to begin softening the broccoli. Add the remaining 2 cups of vegetable stock and bring to a simmer. Let it bubble for about 2-3 more minutes.

Pour the blended rice mixture into the pot. Let it warm through for about a minute or two, stirring frequently, then remove from heat and add the grated cheddar; stir until completely melted. Serve hot with an extra sprinkle of cheese on top if desired or with croutons, crackers or crusty bread.