In a medium bowl, whisk together the eggs, egg whites and sugar; set aside. Gently heat the heavy cream in a saucepan until barely simmering. Remove from heat and add the yeast, stirring until smooth. Stir the cream into the egg mixture and whisk in the flour, followed by 2 tablespoons olive oil. Cover the batter with plastic wrap and refrigerate for 1 hour.
Combine the juice of 1 lemon and 2 cups water in a large bowl. Trim the artichoke stems, snap off the outer leaves until the remaining leaves are half yellow and half green. Cut off the remaining green parts and drop into the lemon water.
Warm 1 tablespoon of olive oil in a medium skillet over high heat. When the pan is hot but not smoking, add the pancetta and cook, stirring frequently, for 1 minute. Drain the artichokes and add them to the pan along with the juice of 1/2 a lemon. Reduce the heat to medium and season with salt and pepper. Cover and cook, stirring occasionally, until the artichokes are tender, 10 to 12 minutes. Transfer to a plate.
Add 1 tablespoon olive oil to the pan. Add the mushrooms and shallot and season with salt and pepper. Cook for 3 minutes, tossing frequently. Add the arugula and 1/4 of the radicchio, season with salt and pepper, and cook, while tossing constantly, until the arugula and radicchio have wilted, about 2 minutes. Transfer the vegetables to a paper towel-lined plate. When cool, coarsely chop the mushroom mixture.
Meanwhile, center a rack in the oven and preheat to 300 degrees F.
Put a 8-inch round cake pan on a baking sheet. Spray the pan with nonstick cooking spray.
Stir the mushroom mixture into the clafoutis batter and season with salt and pepper. Pour half of the batter into the prepared pan and sprinkle the artichoke-pancetta mixture on top. Cover with the remaining batter (the pan should be about 3/4 full). Bake until a knife inserted in the middle comes out clean, 35 to 40 minutes.
Whisk together the remaining lemon juice and remaining 3 tablespoons olive oil. Toss the remaining radicchio with the lemon vinaigrette and season, to taste, with salt and pepper.
To serve, unmold the clafoutis, reinvert right side up, cut into 6 wedges and serve hot, with the radicchio salad.