• Preheat the oven to 375°F. Grease an 8x8-inch baking dish.

For the artichoke filling:

• Using a citrus zester or small-hole grater, zest the lemons. You should get approximately 1 tablespoon per lemon for a total of 4 tablespoons of lemon zest. Set the zest aside for later.
• Squeeze the juice from the lemons; you should have approximately 3/4 cup of lemon juice.
• Into a large pot, combine the water, lemon juice, and sliced artichoke hearts. Bring to a boil then reduce the heat to a simmer for 5 minutes, until the artichokes are softened. Drain and set aside.
• Into a large, deep skillet over medium heat, add the olive oil. When the oil is hot, add the onion slices, 1/4 teaspoon salt, and 1/4 teaspoon pepper. Cook for 8 minutes, stirring frequently, until golden. Remove from the heat.
• To the onion mixture in the skillet, add the artichokes, lemon zest, thyme, parsley, remaining 1/4 teaspoon salt, and remaining 1/4 teaspoon pepper. Stir gently to combine and adjust seasoning to taste.

For the white sauce:

• In a small saucepan over medium heat, melt the butter.
• Whisk in the flour, stirring continuously for 1 minute. The mixture will come together in a thick ball-like shape.
• Gradually add the water and milk, whisking continuously until smooth. Season with salt.
• Reduce the heat to low and continue cooking, stirring frequently until the sauce is thickened and smooth.

Putting it all together:

• Add the sauce to the artichoke mixture, stirring gently to distribute evenly.
• Pour into the prepared baking dish. Cover with foil and bake for 30 minutes. Increase the oven's temperature to 400°F.
• Remove the foil and sprinkle Parmesan cheese over the top. Return to the oven and continue baking for 15 minutes, until the cheese is golden and the sauce is bubbling. Garnish with additional parsley and parmesan cheese then serve hot.