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.