In a small bowl, combine shoyu, pineapple juice, ginger, ketchup, and pepper. Place chicken breasts in a large freezer bag and cover with marinade. Close bag firmly and shake to evenly distribute marinade over chicken. Place bag in refrigerator and marinate 4-5 hours or overnight.
Preheat a gas grill to medium heat.
In a small saucepan, place mango, pineapple, lime juice, and sugar. Heat on medium until mixture begins simmering. Turn heat down to medium-low and let simmer, stirring occasionally, for 15 minutes. Stir in red onion, salt, and pepper flakes and let cook, covered, an additional 5-7 minutes. Remove from heat and set aside.
Brush grill liberally with canola oil. Remove chicken breasts from marinade and place on hot grill. Cover and cook on first side for 8-10 minutes, depending on thickness. Flip and cook an additional 8-9 minutes on the second side, covered. Carefully place one slice of Swiss cheese atop of each chicken breast, cover, and cook for 30 seconds, or until cheese has begun to melt.
Open sandwich buns and place chicken breast on cut-side of bottom bun. Smother with a heaping spoonful of spicy chutney, cover with top bun, and serve.