Place a small saucepan over medium heat and add 3 tablespoons unsalted butter; melt until foamy, about 1 minute, without browning.

Add 3 tablespoons all-purpose flour and stir constantly for about 2 minutes until the roux is pale and slightly toasty.

Reduce heat to medium-low and slowly pour in 2 cups warmed whole milk in three additions, whisking thoroughly between each addition to prevent lumps.

Continue whisking for 3 to 4 minutes until the sauce thickens and coats the back of a spoon.

Lower heat to low and simmer gently for 2 to 3 minutes, stirring occasionally to finish cooking the flour.

Remove the pan from heat and add 1 cup freshly grated Gruyere cheese in handfuls, stirring until each addition melts, about 1 minute.

Stir in 1/2 teaspoon salt, 1/4 teaspoon black pepper, and 1/8 teaspoon nutmeg; taste and adjust seasoning, adding a pinch of cayenne pepper if desired.

Immediately pour the sauce over pasta, vegetables, seafood, or eggs; if it thickens, loosen with a splash of warm milk before serving.