Position rack in lower third of oven and preheat to 400F. Butter 6-cup (1 1/2-quart) soufflé dish or 8 individual ramekins. Add Parmesan cheese and tilt dish, coating bottom and sides.
Warm milk in heavy small saucepan over medium-low heat until steaming.
Meanwhile, melt butter in heavy large saucepan over medium heat. Add flour and whisk until mixture begins to foam, about 3 minutes (do not allow mixture to brown). Remove saucepan from heat; let stand 1 minute. Pour in warm milk, whisking until smooth. Return to heat and cook, whisking constantly until very thick, 2 to 3 minutes.
Remove from heat.
Whisk in paprika, salt, and nutmeg. Add egg yolks 1 at a time, whisking to blend after each addition. Scrape soufflé base into large bowl. Cool to lukewarm.
Can be made 2 hours ahead. Cover and let stand at room temperature. This is good for when you are serving guests, as you can place in oven timed to go right to the table.
Rub the inside of bowl used for beating egg whites with a combination of salt and white vinegar. wipe out excess with a paper towel.
Using electric mixer, beat egg whites in another large bowl until stiff. Fold 1/4 of whites into lukewarm or room temperature soufflé base to lighten. Fold in remaining whites in 2 additions while gradually sprinkling in Gruyère cheese. Transfer batter to prepared dish.
Place dish in oven and immediately reduce oven temperature to 375F. Bake until soufflé is puffed and golden brown on top and center moves only slightly when dish is shaken gently, about 25 minutes (do not open oven door during first 20 minutes).
Serve immediately. Can be a meal served with a hearty salad.