Bring a pot filled with lightly salted water to a boil. Add cauliflower and broccoli and boil until vegetables are tender but firm to the bite, 10 to 15 minutes. Drain well and transfer to a large bowl.

Preheat the oven to 425 degrees F (220 degrees C). Grease a baking dish.

Meanwhile, melt butter in a saucepan over medium heat until a pinch of flour sprinkled into the oil will just begin to bubble. Whisk in flour to form a thick paste and cook and stir, 1 to 2 minutes. Pour in hot milk and continue cooking, whisking constantly, until you have a thick sauce with no lumps, about 5 minutes.

Stir in wine and bring to a boil. Mix in Parmesan cheese and eggs. Season cheese sauce with salt, pepper, and nutmeg.

Remove from heat and mix sauce into the bowl with drained cauliflower and broccoli. Pour mixture into the prepared baking dish and smooth out.

Bake in the preheated oven until gratin is nicely browned on top, 20 to 25 minutes.