Preheat the oven to 425°F. Cut the chicken into 1-inch cubes, slice the pepper, trim the broccoli, shred both cheeses, and chop the parsley.
Spread the broccoli and bell pepper on a parchment-lined baking sheet and roast for 15-18 minutes, turning once, until crisp-tender with browned edges.
Cook the farfalle until one minute shy of al dente. Reserve 1 cup pasta water, then drain without rinsing.
Melt 1 tablespoon butter in a wide skillet over medium-high heat. Cook the chicken with half the black pepper for 7-9 minutes, until browned and 165°F inside, then transfer to a plate.
Lower the heat to medium, melt the remaining 2 tablespoons butter, whisk in the flour, and cook for 1 minute.
Slowly whisk in the milk and simmer gently for 4-6 minutes, stirring often, until the sauce coats the back of a spoon.
Remove the skillet from the heat. Stir in the cheddar and Havarti gradually until smooth, then add the remaining black pepper.
Fold in the pasta, chicken, roasted vegetables, and enough reserved pasta water for a glossy coating. Finish with parsley and serve warm.