Preheat the oven to 350 degrees. Spray a 17- by 11-inch rimmed baking sheet with cooking spray. Line with parchment paper and spray the paper.

Separate the eggs and beat the whites to stiff peaks. Whisk the yolks well.

Heat 2 tablespoons butter in a large skillet over medium heat until melted. Add the flour and cook, stirring, for 1 minute. Add the milk and cook, stirring, until thickened, about 2 minutes. Let cool for 5 minutes. Whisk the egg yolks into the sauce. Stir in 1/3 of the beaten egg whites, then add the mixture to the rest of the egg whites and fold them in. Gently fold in 2/3 cup of Parmesan cheese. Spread the mixture evenly into the prepared pan and bake for 15 minutes.

Meanwhile, whisk the ricotta with the pesto, cream, salt, pepper, and parsley.

When the souffle is done, flip out onto a clean tea towel. Spread the ricotta mixture evenly over the top, leaving a 1-inch border. Roll it, like you would a jelly roll, from the short end, using the towel to help you. Place the roll, seam-side down, on an oven-proof platter. Brush with 1 tablespoon melted butter and sprinkle with the remaining 1/3 cup Parmesan cheese. Bake for 15 minutes. Serve with the warm marinara sauce.