In a Dutch oven over medium heat, brown chicken in 1 Tbsp. olive oil, about 5 minutes.
Meanwhile, stir together ketchup, chili sauce, 1/2 cup honey, water, lime juice and zest, garlic, ginger and pepper. Pour over browned chicken and bring to a boil. Reduce heat to low and cook, covered, stirring occasionally, until chicken is tender, 25-30 minutes. Remove chicken and bring sauce back to a simmer; cook until thick, about 5 minutes. Shred chicken with 2 forks; toss with reduced sauce.
Preheat oven to 375°. Place bottom buns in a greased 13x9-in. baking dish. Place half a slice of cheese on each of the buns; spoon chicken mixture on top of cheese. Top with remaining cheese slices and bun tops. Stir together the remaining olive oil and honey and the pepper flakes. Brush over bun tops. Sprinkle with the coarse salt if desired.
Bake until tops are golden and cheese has melted, 10-15 minutes. Garnish with green onions.
For sauce, stir together all ingredients. Serve alongside sliders.