1. Rinse and drain mushrooms. Trim stems off flush with caps. Trim discolored ends from stems, then finely chop stems. Set caps, cup sides up, on a 12- by 15-inch baking sheet.
2. Cut 4 sheets of foil, each 12 by 16 inches. Fold each sheet lengthwise in half, then in half again. Generously butter 1 side of each foil strip and dust with bread crumbs. Tightly wrap each foil strip, crumb side in, around a mushroom cap, overlapping ends; secure with metal paper clips.
3. In a 2- to 3-quart pan over medium heat, stir 2 tablespoons butter, garlic, and mushroom stems until stems are browned and limp, about 8 minutes. Add flour, salt, and pepper, and stir 1 minute more. Remove from heat and whisk in milk until mixture is smooth. Stir over high heat until boiling, about 1 minute.
4. Remove from heat, add 1 cup cheese, and stir until melted. Add chives and egg yolks, and stir to blend.
5. In a deep bowl with a mixer on high speed, whip egg whites until stiff, moist peaks form. Stir about 1/3 of the whites into cheese mixture, then gently fold cheese mixture into remaining whites just until blended.
6. Spoon mixture equally into mushroom caps. Sprinkle evenly with remaining cheese.
7. Bake in a 375° oven until souffles are well browned, about 30 minutes (about 25 minutes in a convection oven). Quickly remove paper clips and pull foil free, easing away with a knife tip if necessary. With a wide spatula, transfer soufflés to plates.