Preheat the oven to 350 degrees F. Grease a 9 by 13-inch baking dish with 1 tablespoon olive oil.

Bring a medium pot of water to a boil. Add the spinach and cook until just wilted, about 30 seconds. Drain and rinse with cold water. Once cool, squeeze as much water from spinach as possible then roughly chop and reserve.

Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Add the onions and cook until golden brown and tender, about 5 minutes. Add the garlic, 2 teaspoons of Italian seasoning, 1/2 teaspoon salt, and 1/4 teaspoon pepper, and cook, stirring, until the garlic is fragrant, about 30 seconds. Add the artichokes and cook, stirring, another 2 minutes. Remove from the heat and reserve.

Combine the eggs, cream, milk, lemon juice, remaining 1 tablespoon Italian seasoning, remaining 2 teaspoons salt, and remaining 1 teaspoon pepper in a large bowl. Whisk to combine. Add the bread, spinach, artichoke mixture, brie, 1/4 cup Parmesan, and parsley and stir to combine. If bread does not absorb all of liquid immediately then let rest until this happens, about 20 minutes.

Pour the bread pudding mixture into the prepared dish. Sprinkle remaining 1/4 cup Parmesan over the top and drizzle with remaining 2 tablespoons olive oil. Bake until firm in the center and golden brown, about 1 hour. Serve warm.