Cut bread into 1-inch cubes and prepare fruit and vegetable dippers; set aside.

Shred Gruyere, Emmental, and Appenzeller cheeses and toss together.

Sprinkle cornstarch over the shredded cheese and toss until evenly coated.

In a fondue pot or heavy-bottomed saucepan, combine milk and broth and heat over medium heat until gentle simmer (steam rising), about 3 minutes.

Stir in minced garlic and cook for 30 seconds to release flavor.

Reduce heat to medium-low and add cornstarch-coated cheese handfuls at a time, stirring constantly in a figure-eight pattern until smooth, about 3–4 minutes.

Stir in Dijon mustard, lemon juice, nutmeg, and white pepper; season with salt to taste.

Transfer fondue to heated fondue pot and maintain at a gentle simmer (140–160°F); add a splash of hot milk if thickened.