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.
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.
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.
Remove from heat, add 1 cup cheese, and stir until melted. Add chives and egg yolks, and stir to blend.
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.
Spoon mixture equally into mushroom caps. Sprinkle evenly with remaining cheese.
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.