Gather the ingredients.

Melt butter in a high saucepan with a lid over medium-high heat. Add chicken thighs and cook, stirring, until lightly browned, about 5 minutes. Add onion and cook, stirring, until soft and translucent, about 5 minutes. Season with salt, pepper, and cayenne. Mix in flour and stir until flour is well mixed in, making a roux, 2 to 3 minutes.

Pour in chicken broth and scrape up browned bits from the bottom. Add carrots and celery and bring gravy to a simmer. Lower heat to medium-low and maintain a steady simmer. Add thyme leaves and simmer until gravy is slightly reduced and has thickened, about 45 minutes.

Meanwhile, make the dumplings: Combine farmer's cheese, egg, rosemary, salt, and pepper in a bowl. Mix with a spoon until smooth. Add self-rising flour and mix until a very stiff batter forms.

Raise heat to medium and bring gravy back to a steady simmer. Add green onions.

Remove 1 heaping tablespoon dumpling batter at a time. Shape with the use of a second tablespoon into oval dumplings and carefully drop into the simmering liquid. You should get a total of 6 dumplings out of this recipe. Dust with cayenne pepper, cover, and reduce heat to medium-low.

Simmer and steam dumplings until they are puffed up and cooked through without stirring or peeking, about 15 minutes.

Taste and adjust seasoning with salt if necessary.