Melt butter in a large skillet over medium-high heat; add onion, carrot, celery, and garlic, and saute 5 to 6 minutes or until tender. Stir in matzo and next 3 ingredients; let cool.
Add beaten eggs, stirring mixture until well blended.
Shape mixture into a 12-inch roll on a piece of heavy-duty aluminum foil. Fold sides of foil loosely over roll; fold ends over, and crimp to seal. Place on a 15- x 10-inch jellyroll pan.
Bake at 350° for 45 minutes. Unwrap and cut hot stuffing into 1/2-inch-thick slices. Garnish, if desired. Serve immediately.