1. Preheat oven to 350°. Melt butter in a large skillet over medium-high heat; add corn and onion, and sauté 4 minutes. Remove from heat, and cool 10 minutes.
2. Meanwhile, whisk together eggs and next 2 ingredients in a large bowl; gradually whisk in goat cheese until blended. Whisk in basil and next 3 ingredients; stir in corn mixture and 3/4 cup Cheddar cheese until blended. Spoon mixture into a lightly greased 2-qt. baking dish; sprinkle with remaining 1/4 cup Cheddar cheese.
3. Bake at 350° for 40 to 45 minutes or until golden brown and a knife inserted in center comes out clean. Remove from oven to a wire rack, and cool 15 minutes.