Whisk olive oil, honey, lime juice, spices, salt, and pepper. Toss in chicken, coat well, and chill for 30 minutes.
Bring chicken to room temp. Lightly coat with cornstarch. Fry in ¼-inch oil over medium-high heat, 3–4 minutes per side. Drain on paper towels.
Blend avocado, sour cream, lime juice, garlic, cilantro, and salt until smooth. Add water as needed to thin.
Layer tostadas with crispy chicken, drizzle sauce, and top with jalapeños and cilantro. Serve right away.