Pound chicken to even thickness and pat dry.
Whisk pickle brine, milk, 1 tbsp oil, honey, salt, pepper, smoked paprika, garlic powder, and onion powder.
Submerge chicken in marinade. Chill 30 minutes to 4 hours.
Cook bacon in a skillet until crisp. Drain on paper towels.
Stir mayonnaise, barbecue sauce, honey, Dijon, and paprika. Set sauce aside.
Preheat grill to medium-high. Oil grates with remaining 0.5 tbsp oil.
Shake excess marinade off chicken. Grill 4–6 minutes per side, to 165°F.
Top chicken with cheese during last minute. Cover grill to melt.
Spread butter on cut sides of buns. Toast on grill until golden.
Spread sauce on buns. Layer lettuce, tomato, cheesy chicken, and bacon. Cap with top bun.