Preheat oven to 375 degrees F (190 degrees C).
Prick potatoes with a fork and wrap each potato in aluminum foil.
Bake in the preheated oven until potatoes are tender, 1 1/2 to 2 hours. Cool potatoes until easily handled; scoop insides from skins and discard skins.
Place chicken in a bowl and add enough olive oil to coat; add 1 teaspoon salt and 1/2 teaspoon pepper and toss to coat.
Heat a skillet over medium-high heat; cook and stir chicken until no longer pink in the center and juices run clear, 5 to 10 minutes.
Toss onion, carrots, and celery with enough olive oil to coat in a bowl.
Heat a skillet over medium-low heat; cook and stir onion mixture until tender, 5 to 10 minutes.
Melt butter in a large saucepan over medium-heat. Stir flour into melted butter and cook until lightly browned, 1 to 2 minutes. Pour milk, about 1/4 cup at a time, into butter-flour mixture; cook, stirring constantly, until sauce is thickened, about 10 minutes.
Stir cooked potato, chicken, onion mixture, chicken stock, broccoli, 1/2 of the bacon bits, 1/2 of the green onions, and 4 ounces of the Cheddar cheese into the sauce. Cook soup until heated through and broccoli is tender, about 10 minutes; season with salt and pepper. Stir sour cream into soup and cook until heated through, about 5 minutes.
Ladle soup into serving bowls and top with remaining bacon, green onions, and Cheddar cheese.
Soup will continue to thicken as it cools down. If you like your soup really thick, let it cool about 30 minutes before serving.
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