Preheat the oven to 400 degrees F. Grease an 8-by-8-inch baking dish. Place the cooled breadcrumbs in a large bowl. In a large skillet, heat 1 tablespoon oil over medium heat. Add the mushrooms and cook, stirring often, until browned, about 5 minutes; add to the breadcrumbs.
Add 3 tablespoons oil, the onion, salt and pepper to the skillet. Cook over medium heat, stirring often, until the onion is softened and lightly browned, about 8 minutes. Add to the breadcrumb mixture. Add the carrots, celery and baking soda to the bowl and mix well. Stir in the eggs. Spoon the mixture into the prepared dish and spread evenly; bake until browned, 25 to 30 minutes. Let cool for 5 minutes; cut into 16 squares.