Trim roots from onions; discard roots. Chop half of the onions.
Whisk together eggs and milk.
Sift together flour and next 3 ingredients.
Gradually add flour mixture to egg mixture, whisking rapidly 20–30 seconds or just until blended and smooth. (There should be no lumps.)
Stir in chopped onions. Let stand 5 minutes.
Carefully remove hot skillet from oven.
Add butter, and let stand until butter is melted.
Place skillet over medium–high heat (stove top), and pour batter into skillet.
Arrange cheese and remaining whole onions over top of batter, and cook 30 seconds to 1 minute or until edges begin to set.
Transfer skillet to top oven rack, and bake at 400 degrees for 22–25 minutes or until golden brown and puffy. (Outside edges should be crispy and inside texture should resemble a custard popover. Pie will deflate quickly.)