Position rack in lower third of oven and preheat to 400F.Butter 6-cup (1 1/2-quart) soufflé dish. 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 and loses raw taste, 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. DO AHEAD: Can be made 2 hours ahead. Cover and let stand at room temperature.
Using electric mixer, beat egg whites in another large bowl until stiff but not dry. 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 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.