In a large Dutch oven or heavy pot over medium heat, heat oil. Cook onion, carrots, and celery, stirring occasionally, until tender, about 7 minutes. Stir in oregano; season with salt and pepper. Add garlic and cook, stirring, until fragrant, about 1 minute.
Add chicken, broth, and thyme to pot and bring to a boil. Reduce heat to medium-low and simmer, stirring occasionally, until chicken is cooked through, about 10 minutes. Transfer chicken to a plate or cutting board and shred with two forks. Keep soup at a simmer until ready to use.
In a large bowl, whisk flour, baking powder, and salt. Make a well in center of flour mixture. Add egg, buttermilk, and butter to well and whisk with a fork until combined.
Add shredded chicken and cream to soup, then return to a simmer. Drop spoonfuls of dumpling mixture into pot. Cover and cook over low heat until dumplings are cooked through, about 5 minutes.
Top soup with parsley and divide among bowls.
