Slice 2 1/2 pounds small potatoes as directed (see left); transfer to a large bowl and add 1 large red onion (cut into wedges). Melt 6 tablespoons butter in a small skillet over medium-high heat, swirling, until browned, 2 to 3 minutes; pour into a small bowl. Whisk in 1 1/2 teaspoons smoked paprika, 1 teaspoon kosher salt and a few grinds of pepper. Drizzle half of the brown butter over the vegetables; toss to coat. Spread on a rimmed baking sheet and roast at 425 degrees F until tender, 35 to 40 minutes. Drizzle with the remaining brown butter and roast until golden brown, 30 more minutes. Sprinkle with shredded manchego cheese, chopped chives and flaky sea salt.
Cut a thin sliver from one side of the vegetable so it can sit flat. (If you¿re starting with halved veggies, like the beets at right, skip this step.)
Position the vegetable cut-side down between the handles of two wooden spoons.
Using a sharp knife, make crosswise cuts (1/8 to 1/4 inch apart), stopping when the knife hits the handles so you don't cut all the way through the vegetable.