Fill a medium pot a third of the way with water, cover with a heat-proof bowl, and bring to a boil over medium-high heat.

In a separate large bowl, toss together the Gruyère and Swiss cheeses; add the cornstarch and toss to coat.

To the heated bowl, add the pinot gris. When the wine is steaming, add a handful of cheese and whisk continuously until melted before adding another handful.

Continue until all the cheese is melted and the mixture is silky smooth. Finish with the lemon juice and season with salt and pepper to taste.

Transfer the dip to a fondue pot and serve immediately with long toothpicks or fondue forks for dipping.