To make the tostadas, line a baking sheet with paper towels. Place 4 Tbsp. of the oil in a skillet set over medium, medium-high heat. Cook the tortillas, one at a time, until golden and crispy, about 30 seconds per side. Transfer each cooked tortilla to the paper towels to drain. When all are cooked and cooled, cover the tostadas with plastic wrap and set aside at room temperature until ready to top.
To make the chicken tinga, heat the remaining 2 Tbsp. of oil in a medium pot set over medium, medium-high heat. Add the onion and cook until tender, about 4 minutes. Stir in the tomatoes, cumin and oregano and cook and stir 2 minutes more. Mix in the salsa verde, stock and sugar. Bring to a simmer and cook 5 minutes. Add the chicken to the pot, return to a simmer and cook 5 minutes more.
To serve, place the queso fresco, avocado, jalapenos, sour cream and cilantro in individual small bowls. Spoon the chicken tinga mixture into a heatproof serving dish. Set all, including the tostadas, on the dinner table. Let diners top their tostadas to taste, with the chicken tinga mixture and other ingredients.