Toss fontina and asiago cheese together with cornstarch.
Melt butter in a large sauce pan over medium heat; add spinach, garlic, red pepper flakes, anchovy paste, and nutmeg; cook until garlic softens, about 1 minute.
Add wine and broth; bring to a simmer. Whisk in cheese mixture, a handful at a time, letting each handful melt fully before adding more. Whisk in cream cheese cubes and stir until cubes melt.
Stir in artichokes, and lemon juice; season to taste with salt and pepper, then transfer to a warm fondue pot and serve with assorted breads, vegetables, sliced apples, pretzels, and crackers.
Makes approx. 3 cups.