Make tostaditas: Heat 1/4 in. oil in a large frying pan over medium-high heat until shimmering. Cook 4 tortillas at a time, turning once or twice, until deep golden and crisp, 2 to 3 minutes. Transfer to paper towels; set aside.
Spoon 1 tbsp. hot oil into a second large frying pan over medium-high heat. Sauté onion a few minutes until golden, then add garlic, salt, and pepper and cook until garlic starts to brown, 30 to 60 seconds. Stir in 3/4 cup water, the tomato paste, chipotle purée, and sugar; add chicken. Cook, stirring, until bubbling, about 2 minutes. Stir in more water if needed to make it saucy.
Set tortillas on plates. Top with chicken. Garnish with cabbage, salsa, queso fresco, cilantro, and avocado. Drizzle with crema.
*Find canned chipotle chiles, queso fresco, and crema at Latino markets. Purée contents of chipotle can in a blender; keeps, chilled, up to 1 month.
Note: Nutritional data is per tostadita.