Bring 4 qt. water to a boil in a 6-qt. Dutch oven.
While water comes to a boil, remove and discard stems and ribs from chard. Coarsely chop leaves. Wash chard thoroughly in cold water; drain well. Process eggs and next 5 ingredients in a blender until smooth.
Add chard leaves and pasta to boiling water. Return water to a boil; boil 1 minute. Drain well, pressing out excess moisture from chard with the back of a spoon. Return pasta and greens to pan; toss until blended.
Brush a 10" cast-iron skillet with olive oil. Arrange pasta and greens in skillet; sprinkle with cheese. Pour egg mixture over cheese, pressing to submerge pasta and greens in liquid.
Place skillet over medium heat for 2 minutes. Immediately transfer to hot oven. Bake at 400° for 25 minutes or until golden and set. Let stand 10 minutes before inverting onto a serving platter, if desired, and cutting into wedges.
*It's easier to prepare greens if you chop or tear them first before cleaning in several washings of cold water. Drain well between washings, and finish with a whirl in a salad spinner to remove sandy grit.