To prepare chutney, combine first 9 ingredients in a medium saucepan; bring to a boil. Reduce heat, and simmer 15 minutes or until most of liquid evaporates and mixture is thick, stirring occasionally.
To prepare chicken, heat oil in a large nonstick skillet over medium-high heat. Add garlic and chicken; sauté 5 minutes. Combine coconut milk and next 4 ingredients (coconut milk through fish sauce), stirring with a whisk. Add coconut mixture to pan; bring to a simmer. Add broccoli and cauliflower; cover and cook 7 minutes or until vegetables are crisp-tender, stirring occasionally. Stir in chutney and pasta; toss well to combine. Sprinkle with green onions, if desired.