Preheat your oven to 350°F (175°C) and generously butter a 1½-quart baking dish. Slice your Vidalia onions into thin, uniform slices about ⅛-inch thickness.
In a large skillet, melt the 3 tablespoons of butter over medium heat. Once melted, add the sliced onions, stirring to coat them in butter.
Cook for 15-20 minutes, stirring occasionally, until the onions become translucent and softened but not browned.
While the onions are cooking, whisk the eggs in a large bowl until lightly beaten. Add the half-and-half, flour, sugar, salt, pepper, and nutmeg. Whisk until the mixture is smooth and the flour is completely incorporated without any lumps.
Allow the cooked onions to cool for about 5 minutes, then gradually stir them into the egg mixture.
Pour this combined mixture into your prepared baking dish. If using, sprinkle the Parmesan cheese evenly over the top.
Bake in the preheated oven for 50-55 minutes, or until the center is set and the top is golden brown. The pudding should have a slight wobble in the center when gently shaken.
Allow the pudding to rest for 10 minutes before serving. Just before serving, sprinkle with freshly chopped chives or green onion tops.