Set a rimmed baking sheet on the top oven rack; preheat to 475 degrees F. Put the potatoes and 1 tablespoon water in a large microwave-safe bowl; cover with plastic wrap and pierce a few times with a knife to vent. Microwave until very tender, 10 to 13 minutes. Meanwhile, generously butter an 8-inch-round cake pan; sprinkle in 2 tablespoons breadcrumbs and shake to coat, then tap out the excess.
Add 4 tablespoons butter and the parmesan to the potatoes; season with salt and pepper and mash. Gradually mash in the beaten eggs. Spread half of the mixture in the prepared pan; top with the provolone, peppers and parsley, leaving a 1-inch border. Top with the remaining potato mixture and smooth evenly. Sprinkle with the remaining 1 tablespoon breadcrumbs. Dot with the remaining 1 tablespoon butter.
Set the pan on the hot baking sheet and bake until puffed and golden, about 15 minutes. Run a knife around the edge to loosen, then unmold. Let rest 5 minutes before slicing. Meanwhile, toss the salad greens, olive oil and lemon juice in a bowl; season with salt and pepper. Serve with the potato cake.