Heat 1 tablespoon olive oil over medium-high heat in a large skillet or cast iron pan. Add prosciutto and sauté until beginning to cook, 1-2 minutes.
Add onion and sauté until prosciutto is crisp and onion is golden, stirring several times, about 4 minutes more. Add chard stems and garlic and cook an additional 30 seconds.
In batches, begin adding the chard and kale leaves, rotating them with a spatula as the leaves on the bottom begin to wilt. Continue to add more leave, carefully tossing so that all the leaves begin to wilt.
Add half and half, nutmeg, and salt and pepper to taste, and simmer a few minutes more.
Remove from heat and sprinkle freshly-grated cheese over the top.