In 4-qt stock pot or Dutch oven, melt butter over medium heat. Whisk in flour and simmer, whisking constantly, until smooth and bubbly. Whisk in cream, milk, and chicken bouillon and cook, stirring frequently or whisking, over medium-low heat until smooth and thickened.

Heat oil in a large saute pan over medium heat. Add onion and cook until soft, about 3 minutes. Add garlic and carrots; cook until tender, about 3 minutes, stirring frequently. Add poblano peppers and cook until tender, about 3 minutes.

Add vegetable mixture to milk mixture; stir to combine. Stir in salt and ground black, white, and red peppers; simmer over medium-low heat a few minutes.

Add red bell pepper. Simmer mixture 15 to 20 minutes, stirring frequently.

Add cheese and stir until melted. Taste and adjust seasonings, if necessary. Serve poblano soup hot or refrigerate immediately and reheat to serve.