Preheat the oven to 425 degrees F. Prick the potatoes all over with a fork and place on a baking sheet. Bake, flipping halfway through, until fork-tender, about 45 minutes. When the potatoes are cool enough to handle, cut them in half and carefully scoop the flesh into a large bowl. Add the flour and mash with a fork until just a few chunks of potatoes remain. Make a well in the middle of the mixture and set the bowl aside. Reserve the baking sheet.
Meanwhile, fill a medium saucepan with 2 inches of water. Set a steamer basket insert into the pot and bring the water to a simmer over medium-high heat. Add the cauliflower, cover and steam until very tender and falling apart when pierced with a fork, 12 to 14 minutes. Drain the cauliflower and use clean kitchen towels to squeeze completely dry.
Add the cauliflower a food processor along with the Parmesan, egg yolk, 1 teaspoon salt and a few grinds of black pepper. Puree until the mixture is smooth. Pour the puree into the well of the potato mixture and use a fork to gently mix the potatoes and cauliflower until just combined and a soft dough is formed. (Do not overmix the dough or the gnocchi will be unpleasantly dense.)
Generously flour a work surface. Scoop a heaping 1/2 cup of dough onto the surface and roll into a 12-inch rope. Use a knife to cut the dough into 1-inch pieces and place them on the reserved baking sheet. Repeat with the remaining dough. Gently press the top of each gnocchi with a fork to make grooves.
Melt 1 tablespoon butter in a small microwave-safe bowl in the microwave. Brush the gnocchi with the butter. Bake the gnocchi until cooked through and they spring back when pressed, 10 to 12 minutes.
Melt the remaining 4 tablespoons butter in a large skillet over medium heat until brown bits start to form, 5 to 6 minutes. Remove from the heat, add the capers and stir until well combined. Toss the gnocchi in the brown butter and sprinkle with the parsley. Serve immediately with a salad on the side.
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