Preheat the oven to 375°F (190°C) and butter a 12-cup (3-L) baking dish or soufflé dish. Cut the bread into ¼-inch (.6-cm) cubes and place in a bowl large enough to hold all the ingredients. Toss the cheeses with the bread.
Put 2 whole eggs and 4 egg yolks in a bowl. Reserve the 4 leftover egg whites in another bowl.
Whisking, add the milk, mustard, chives, salt, and pepper to the whole eggs and egg yolks. Pour over the bread and cheese mixture and mix gently. Let stand at least half an hour. Whisk the reserved egg whites with a pinch of salt until they hold stiff peaks. Stir a third of the whites thoroughly into the bread mixture, and then gently fold in the rest.
Pour the mixed ingredients into the baking dish. If using a shallow baking dish, cook for 45 minutes. A higher soufflé dish will need to bake for about 1 hour and 15 minutes. The top should become a dark golden brown. Cover during baking if it starts to burn. Serve immediately. If you want, garnish each serving with sour cream and chives. The soufflé may sink in the middle, but that won't affect the taste.