Heat oil in a Dutch oven over medium heat until shimmering. Add the onion and cook until softened and lightly browned, 5 to 7 minutes. Stir in the tomato paste and thyme, cooking until fragrant, about 30 seconds.
Stir in chicken broth and salt, scraping up any browned bits. Add carrots and celery, then bring the mixture to a simmer. Nestle the chicken breasts into the pot. Reduce heat to low, cover, and cook until the chicken registers 160°F, about 10 minutes.
Remove the chicken to a cutting board, let it cool slightly, and shred it into bite-sized pieces using two forks.
Meanwhile, bring the soup back to a boil. Stir in the egg noodles and cook until tender, 5 to 8 minutes.
Off the heat, stir the shredded chicken back into the soup. Let it sit until heated through, about 2 minutes. Stir in parsley, then season with salt and pepper to taste. Serve warm.