Combine the carrots, peppers, onions, and cabbage in a large bowl.
In a medium saucepan, add the vinegar, water, sugar, salt, oregano, crushed pepper, coriander, and cumin, and bring to a boil. Pour the hot pickling liquid over the vegetables. Mix the ingredients well and let stand for at least 24 hours before serving, stirring occasionally.
Season the turkey with the salt and pepper, and place in a medium saucepan set over medium heat.
Meanwhile, combine the tomatoes, pepper, onions, oregano, and cinnamon in a blender and purée until smooth. Add the tomato mixture to the turkey in the saucepan and stir to loosen any browned bits from the bottom of the pan. Lower the heat and simmer for another 15 minutes, or until turkey is falling apart and most of the liquid has reduced from the sauce. The turkey mixture should be very thick. Set aside to cool while you prepare the masa dough.
In a bowl, combine the masa flour with the warm water and salt. Mix the ingredients until a soft dough forms. Let the dough stand for 20 to 30 minutes.
Stir in about ¼ cup cold water until the dough become soft but not sticky. Divide the dough into 16 golf-ball-size pieces. Keep the dough covered as you work.
Rub the palm of your hands with a little olive oil. While holding of the dough balls in 1 hand, press the thumb of the other hand in the middle of the ball of dough to form an indentation, then turning the dough, begin to flatten it with your fingertips into a 4-inch concave disk resembling a small bowl. In the center of the disk add 2 tablespoons of the turkey mix and 1/2 tablespoon of the shredded cheese. Bring the edges of the dough together over the filling and squeeze to form a seal. Working gently, presses the dough into a flattened disk, flattening the dough middle and edges to form a thick pancake shape about 4-inches in diameter. Repeat with the remaining dough and filling.
Lightly rub the pupusas with oil and place them on a heated, greased griddle over medium-high heat. Cook each side until golden brown in spots and slightly puffy, 3 to 4 minutes per side. Serve hot, with a serving of the pickled cabbage on the side.