Heat 3 tablespoons oil in a Dutch oven or large cast iron skillet over medium heat. Pat chicken dry with a paper towel and season to taste with all-purpose seasoning.
Cook chicken thighs in the hot oil until browned, 4 to 5 minutes. Transfer chicken to a platter to cool.
Add onions, celery, carrots, and garlic to the remaining oil in the Dutch oven (add 1 tablespoon more oil if needed). Season mixture with 1 teaspoon all-purpose seasoning to layer the flavor. Cook and stir until onions begin to turn translucent, 3 to 4 minutes.
Add mushrooms, chicken bouillon cube, and dried rosemary. Cook and stir for 2 minutes. Whisk in flour; cook for 1 to 2 minutes.
Slowly whisk in chicken stock. Bring to a simmer and continue stirring until flour is well mixed into stock. Cook, stirring constantly, until soup begins to thicken. Depending on how thick you would like your soup, you may need to add a little more stock.
Meanwhile, remove skin from chicken thighs, pull the meat apart, and add to the pot. Continue to stir for 1 minute. Cover and cook at a low simmer for 10 to 12 minutes.
Add frozen vegetables; stir and reduce heat. Pour in heavy cream and stir well to blend in the cream. Cover, and return soup to a simmer, 3 to 5 minutes. Remove from heat, stir, and serve immediately.