Heat olive oil in a medium iron skillet over medium heat. Place potatoes in the skillet, cover, and cook about 10 minutes, until tender but firm. Mix in peppers and garlic. Season with salt and pepper. Add basil and cook an additional 1 to 2 minutes
In a medium bowl, beat together eggs and cream. Pour into the skillet over the vegetables. Sprinkle with goat cheese. Reduce heat to low, cover, and cook 5 minutes.
Finish under broiler for 2 minutes or until top has slightly browned.