Preheat oven to 375°F. Heat oil and 2 tablespoons butter in a large nonstick skillet over high until butter melts. Stir in celery and leeks; cook, stirring occasionally, until tender, about 5 minutes. Add garlic; cook, stirring, 1 minute. Add thyme, salt, pepper, and 1/2 cup of the broth; cook, stirring, 1 minute.
Combine celery mixture and bread cubes in a large bowl. Add melted butter, eggs, and remaining 1 1/2 cups broth; stir to combine.
Spoon mixture into a lightly greased 11- x 7-inch baking dish; let stand 15 minutes. Bake in preheated oven until golden brown, 40 to 45 minutes. Sprinkle with parsley.
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