Preheat oven to 375 degrees F (190 degrees C). Grease a 9-inch pie pan.
Heat oil in a saucepan over medium-high heat. Saute onions in the hot oil until softened and lightly browned, about 5 minutes. Add celery and cook until soft, about 5 minutes; stir in chicken broth, garlic, parsley, sage, rosemary, and thyme. Simmer broth mixture until it thickens slightly, about 10 minutes. Remove saucepan from heat and pour broth mixture into a bowl. Set it aside to cool, about 10 minutes.
Stir eggs and feta cheese into broth mixture; pour into the prepared pan.
Bake in the preheated oven until set, 25 to 35 minutes.