Heat a large saucepan over medium-high heat. Add the bacon and cook, stirring often, until the bacon crisps and the fat is rendered, about 7 minutes. Using a slotted spoon, transfer the bacon to a paper towel-lined plate. Discard all but 2 tablespoons of bacon fat.
Add the onion, bell pepper, jalapeño, and thyme to the reserved bacon fat and cook, stirring often, until softened, about 5 minutes.
Reduce the heat to medium. Add the butter and cook, stirring, until completely melted. Sprinkle the flour over the mixture and stir until the flour is fully incorporated. Once the flour is incorporated, add the beer and cook, stirring often, until the mixture thickens, about 5 minutes. Gradually stir in the chicken stock and cream until thoroughly combined and bring to a low simmer.
Once the mixture is simmering, add in the cheeses a handful at a time, stirring until melted before adding more, about 5 minutes total. Stir in half of the bacon. Season with salt and pepper to taste.
Garnish with the remaining bacon, and green onions before serving.