Heat olive oil over medium-high heat and cook finely chopped onion and finely shredded carrots. Season with salt and pepper. You can add 1 - 2 Tbsp water to the skillet and cover with a lid. Cook vegetables until they are tender.
In the large mixing bowl add mashed potatoes, cooked onion, carrots, crumbled feta cheese and shredded cheese and season with chubritsa or Italian seasoning, salt, and pepper.
Whisk the 2 eggs SEPARATELY in two small bowls. Add 1 egg to the potato mixture and mix well. Form small potato patties, dip in the second whisked egg and then dredge in the gluten free flour.
Heat olive oil over medium heat. Working in batches, fry potato patties in the hot oil, until golden brown, and turn the patties on the other side. Add olive oil as needed for each new batch.
Place cooked patties on a plate and serve warm. You can serve these patties with the yogurt mayonnaise garlic dip from this recipe.