Grate the cheeses using a box grater or an attachment for your food processor. Add grated cheeses to a large bowl and toss in the cornstarch. Mix well, coating all the pieces of shredded cheese.
Add the wine and lemon juice, and bring to a simmer over medium heat. Stir in a handful of the shredded cheese and gently mix until the cheese is melted.
Continue adding the cheese a handful at a time, letting the cheese melt before adding more to ensure a smooth fondue. Once smooth, stir in the dijon mustard, pepper, and nutmeg.
Serve immediately, or transfer to a fondue pot, along with an assortment of cheese fondue dippers such as pretzels, baguette rounds, cherry tomatoes, broccoli, apples, and grapes. Enjoy!