In a large skillet over medium heat, warm the oil. Add the bacon and fry until golden and crisp, 5 to 10 minutes. Use a slotted spoon to scoop out the bacon and transfer it to a paper towel-lined plate, leaving the fat in the skillet.
Season the chicken with salt and pepper and fry in the same skillet over medium-high heat until cooked through to an internal temperature of 165°F (74°C), 3 to 4 minutes per side.
Return the bacon to the skillet and add the butter and garlic. Once the butter has melted, add the wine, mustard, honey, capers, hot sauce, and half of the parsley and let reduce to a thickened sauce, 2 to 4 minutes. Season with salt and pepper.
Meanwhile, bring a large pot of water to a boil over high heat. Toss in the broccolini and blanch for 2 to 3 minutes, then drain.
Serve the chicken with the broccolini, scattered with the remaining parsley.