Heat 1 tablespoon oil in a skillet over medium heat. Add onion, artichokes, and thyme; cook 5 minutes.
Heat a cast-iron skillet over medium-high heat. Add remaining 3 tablespoons oil and potatoes; cook 4 minutes. Stir in onion mixture, bell peppers, 1 tablespoon oregano, salt, black pepper, chicken, half of cheese, and garlic cream. Cook 2 minutes. Stir in half-and-half.
Divide hash among 6 plates. Sprinkle with remaining 1 tablespoon oregano and cheese.