In a Dutch oven, heat vegetable oil over medium heat. Season chicken breasts with 1½ teaspoons salt. Sear chicken until golden, about 5 minutes per side. Remove chicken and set aside.
Meanwhile, cut carrots crosswise into ¼-inch-thick half-moon slices.
Add butter and melt. Add flour and cook, whisking frequently, for 2 minutes. Add onion, carrots, celery, and garlic; cook until vegetables begin to soften, about 5 minutes. Add mushrooms and stir to combine.
Add chicken broth, thyme sprigs, and remaining 1½ teaspoons salt; bring to a boil. Add wild rice and return chicken to pan. Reduce to a simmer and cook until chicken and rice are tender, about 45 minutes.
Remove thyme sprigs. Shred chicken breasts and return to pot. Add cream and cook until hot and thickened, 5 to 10 minutes. Stir in parsley and crushed red pepper. Garnish with parsley, if desired.