Generously season chicken cubes with 3/4 teaspoon salt and 1/2 teaspoon pepper.
Heat oil in a 5-quart (or larger) heavy bottom pot, over medium-high heat, until oil shimmers. Add chicken cubes to shimmering oil and cook, tossing frequently, to brown on all sides. Transfer cooked chicken to a plate; set aside.
In the pot, add more oil if needed, stir in onion (1 cup), carrots (2), and celery (3 stalks) then sauté, over medium-low heat, until soft, about 6-8 minutes. Reduce the heat to low, and add in garlic (1 tablespoon), stirring often, and continue to sauté until garlic is fragrant, 1 minute.
Add dissolved Better Than Bouillon (1 heaping tablespoon) with water and 4 cups of chicken broth. Stir with a wooden spoon, digging down to the bottom of the pot scraping to loosen up all the brown bits off the bottom.
Add fresh thyme (2 sprigs), 1/2 teaspoon salt, 1/2 teaspoon Cajun seasoning, 1/4 teaspoon black pepper, pinch of sugar.
Stir in cubed russets and bring to a boil. Reduce heat to medium-low to low and gently simmer until the potatoes are almost done.
Stir in browned chicken chunks, along with any juices that may have accumulated on the plate, and continue to simmer until the chicken is fully cooked through and the edges of the potatoes are breaking off. Adjust seasoning, adding more salt and pepper as needed. Ladle into individual bowls and serve.