In a large pot over medium heat, cook bacon, stirring occasionally, until crisp, 8 to 10 minutes. Transfer bacon to a plate, leaving about ¼ cup bacon fat in pot.
Add onion, bell pepper, chopped jalapeños, garlic, and ½ teaspoon salt to pot and cook over medium heat, stirring, until softened, 5 to 6 minutes.
Stir in flour until it coats vegetables and forms a paste that starts to stick to bottom of pot. Whisk in broth and milk until combined. Bring to a boil, then reduce heat to medium-low and simmer, stirring occasionally, until liquid is reduced by about one-quarter and slightly thickened, about 10 minutes.
Stir in cream cheese until fully melted, then add chicken and three-quarters of bacon. Cook, stirring, until chicken and bacon are heated through, about 1 minute. Taste and season with remaining ½ teaspoon salt, if needed.
Divide soup among bowls. Top with cheddar, remaining bacon, and sliced jalapeños as desired.