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In a small microwave-safe bowl, melt 3 tablespoons of the butter; set aside.
If the baked potatoes have been refrigerated or are not hot, wrap in paper towels, removing any foil, and heat each one separately the microwave for 45 seconds, or until the potatoes are piping hot and you need to handle them with a kitchen towel.
Cut the potatoes in half lengthwise and scoop out as much of the potato "flesh" into a large bowl as you can while keeping the skins intact as much as possible; set the skins aside.
Process the potato flesh through a ricer or food mill, or if you like rustic-style potatoes, roughly mash them. (Don't use a food processor or you will get gummy potatoes). Place the potatoes back in the bowl, add about half the melted butter (save the other half for drizzling at the end), 1 teaspoon of the salt, and 1/2 teaspoon of the pepper. Mix gently, just to incorporate. Do not overwork or your potatoes will become gluey! Set aside.
In a medium saucepan, heat the remaining 3 tablespoons butter over medium-high until it melts and begins to foam. Add the flour, whisk to combine, and cook until the mixture becomes a blonde color, about 2 minutes. Add the milk, whisk until smooth, and cook until heated through. Add the sour cream, whisk to combine, and cook until heated through. Add the Parmesan cheese and 1 1/4 cups of the cheddar cheese and stir until the cheese is melted and incorporated. Remove the pan from the heat.
Arrange a rack in the middle of the oven and heat to 385°F. Coat a 9x13-inch baking dish with cooking spray.
Transfer half of the potato flesh into the baking dish and spread into an even layer. Arrange the broccoli evenly on top. Pour or scoop the cheese sauce over the broccoli to coat. Cover with the remaining potato flesh in an even layer. Top with the reserved potato skins, skin-side up.
Drizzle the potato skins with the remaining reserved melted butter, then spray them with cooking spray just to coat. Sprinkle with the remaining 1/2 teaspoon salt, half of the chives, and remaining 1 1/4 cups cheddar. Bake until the cheese melts and the casserole is completely warmed through, 10 to 15 minutes.
Top with the crumbled bacon, red pepper flakes, and remaining chives. Serve immediately.