Season the chicken pieces with 1 teaspoon salt and 1 teaspoon pepper.
In a bowl, whisk together 3 tablespoons olive oil, 3 tablespoons honey, 2 teaspoons grated fresh ginger and 2 minced garlic cloves until combined.
Place the chicken in a baking dish or resealable bag and pour the marinade over it. Refrigerate and marinate for 2 to 24 hours.
When ready to cook, heat a large nonstick skillet over medium heat. Add the chicken (discard any excess marinade), cover and cook, stirring once or twice, until golden and cooked through, about 6 to 8 minutes.
After cooking, shred or chop the chicken slightly if desired.
To make the pomegranate-lime salsa, combine 1/2 cup pomegranate arils, 1/4 cup chopped cilantro, 1/2 diced red onion, the sliced green onions, juice of 2 limes, 1 tablespoon pomegranate juice and a pinch of salt and pepper in a bowl; stir to combine.
To crisp the tortillas, heat a large skillet over medium heat and add the vegetable oil a tablespoon at a time. Fry each 4-inch tortilla, flipping once, until golden and bubbly on both sides, about 1–2 minutes per side. Drain on paper towels.
Assemble the tostadas: place shredded chicken on each crispy tortilla, sprinkle with sliced green onions, and top with the pomegranate-lime salsa.