Position a rack in the upper third of the oven and preheat to 400 degrees . Grease a 9-by-13-inch baking dish. Using your hands, crush the saltines into fine crumbs over a bowl.
In a large skillet, melt 1 tablespoon butter over medium heat. Add the mushrooms and cook, stirring often, until browned, about 5 minutes; add to the ground saltines.
Add 3 tablespoons butter, the onion, salt and pepper to the skillet and cook, stirring often, until the onion is softened and lightly browned, about 8 minutes. Add to the mushroom-saltine mixture. Add the carrots, celery and baking soda; mix well. Stir in the eggs.
Spoon the mixture into the prepared dish and spread evenly. Melt the remaining 1 tablespoon butter and brush on top of the mixture. Bake until browned, 25 to 30 minutes. Cut the stuffing into squares and serve hot.