In a large saucepan, heat the oil over medium-high heat. Add the onion and celery and cook until translucent. Add the carrot, corn kernels, garlic, and jalapeno. Cook for 5 minutes. Add the ground turkey and saute until cooked through. Stir in the tomato puree, bay leaf, thyme, oregano and salt and pepper, to taste. Reduce the heat to medium, cover the pan and cook for 10 minutes. Discard the bay leaf.
Put about 1/4 cup of the picadillo into each hamburger bun. Serve warm with pickled jalapenos and avocado slices.