Transfer mixture to small bowl.DO AHEAD: Can be made 2 days ahead.Cover and refrigerate.
Coat inside of 14-cup soufflé dish(about 8 1/4 inches wide and 3 3/4 inchesdeep) with remaining 2 tablespoons butter.
Sprinkle dish with 1/4 cup sugar and tilt tocoat bottom and sides evenly.
Whisk 1/4 cup milk, egg yolks, 4tablespoons sugar, cornstarch, and salt inmedium bowl to blend. Bring remaining13/4 cups milk and 4 tablespoons sugar tosimmer in heavy large saucepan. Graduallywhisk hot milk mixture into yolk mixture.Return custard to same pan. Stir overmedium heat until custard thickens andboils, about 3 minutes.
Remove from heat.
Add chestnut paste, Armagnac, and vanillaand whisk to blend well (some small piecesof chestnut paste will remain). DO AHEAD: Soufflé base can be made 2 hours ahead.Press plastic wrap onto surface; let stand atroom temperature.
Position rack just below center of ovenand preheat to 400°F. Beat egg whitesand cream of tartar in large bowl until softpeaks form. Gradually add remaining 6tablespoons sugar, beating until stiff butnot dry. Fold whites into soufflé base in3 additions.
Transfer batter to prepareddish.
Place souffléon small baking sheet.DO AHEAD: Can be made 1 hour ahead; letstand at room temperature.
Bake soufflé until puffed and just firmto touch in center, about 50 minutes.
Serveimmediately with sauce.
* Peeled cooked chestnuts; sold at somesupermarkets and at specialty foods stores.
To serve the soufflé,use a very large spoon to gently scoop upsome of the saucy interior along with a bitof the firmer sides.