1. In 10-inch skillet over high heat, heat carrots and 1/2 inch water to boiling. Reduce heat to low; cover and simmer 5 to 7 minutes, until carrots are tender-crisp. With slotted spoon, remove carrots to large platter. In same water in skillet, cook broccoli about 10 minutes; remove to platter. Then cook peppers about 2 minutes; remove to platter.
2. To same skillet, add potatoes and enough water to come up to 1 inch; over high heat, heat to boiling. Reduce heat to low; cover and simmer 15 minutes. Remove potatoes to platter with vegetables. Discard water in skillet; wipe dry with paper towels.
3. In same skillet over medium heat, in hot salad oil, cook garlic until lightly browned; discard garlic. Reduce heat to low; add chili powder; cook 1 minute. Stir in milk, Worcestershire, hot-pepper sauce, American cheese, and chopped green chilies until smooth, stirring