Marinade:
In a large bowl, stir together oil, soy sauce, brown sugar, and garlic. Add chicken and toss to coat. Cover and place in refrigerator for about 1 hour.
Preheat grill to high. Lightly oil the grates. Grill chicken until cooked thoroughly (165°F internal temperature).
Salad:
Melt butter in skillet over medium heat. Add crushed ramen noodles, almonds, and sesame seeds. Cook, stirring occasionally until toasted (about 5 minutes). Remove from heat and let cool.
In a saucepan, combine oil, sugar, and vinegar. Bring to a boil and cook for 1 minute. Remove from heat and let cool. Stir in soy sauce.
Chop chicken into bite sized pieces and place in a large serving bowl with cabbage and green onions. Add noodle mixture and soy sauce mixture; toss to coat.