Sprinkle the chicken with 1/2 teaspoon salt and a few grinds of pepper. Heat the oil and 2 tablespoons of the butter in a large pot or Dutch oven over medium-high heat. Add the chicken, and cook until browned in spots but not cooked through, about 2 minutes per side. Remove to a plate; set aside. Add the mushrooms, carrots, leeks, 1/2 teaspoon salt and a few grinds of pepper to the pot, and cook, stirring, until all the vegetables are tender, about 6 minutes. Sprinkle in 3 tablespoons of the flour and stir until absorbed. Add the chicken broth and 2 cups of water, and scrape up any brown bits on the bottom of the pot. Bring to a simmer. Add the chicken, cauliflower and thyme; return to a simmer and cook, stirring occasionally, until the chicken is cooked through and the sauce is about the consistency of heavy cream, about 15 minutes. Discard the thyme. Remove the chicken and use 2 forks to shred the meat. Stir the shredded chicken back into the sauce, along with 1 cup of the Cheddar. Continue to stir until the Cheddar has completely melted. Season with 3/4 teaspoon salt and a few grinds of pepper. While the sauce is simmering, combine the remaining 3/4 cup flour, 1/2 cup of the remaining Cheddar, 2 tablespoons of the chives, the baking soda, 1/2 teaspoon salt and a few grinds of pepper in a small bowl. Break up the remaining 2 tablespoons butter into small pieces, and work into the flour mixture with your fingers until it's completely absorbed. Add the milk, and stir to combine into a wet dough. Drop tablespoonfuls of the dough all over the surface of the sauce. Bring to a low simmer, cover and cook until the dumplings are cooked through, 12 to 15 minutes. Sprinkle the remaining 1 tablespoon chives and some of the remaining 1/2 cup Cheddar on top, and serve family style or in individual bowls.