Place a rack in top third of oven; preheat to 425°. On a baking sheet, toss broccolini and 2 tablespoons oil; season with salt and pepper. Spread in an even layer.
Roast broccolini until beginning to soften, about 10 minutes.
Meanwhile, in a medium bowl, gently toss Halloumi, paprika, oregano, garlic powder, and 2 tablespoons oil until cheese is coated.
Move broccolini to edges of baking sheet and arrange Halloumi slices about ¼" apart in center of sheet. Continue to roast until broccolini is charred and Halloumi is browned and crisp, 15 to 20 minutes more.
Meanwhile, in a food processor, pulse chives, parsley, garlic, lemon zest, and lemon juice, scraping down sides of bowl as needed, until mostly smooth. With the motor running, slowly pour in remaining ½ cup oil, scraping down sides, until mixture is emulsified. Add walnuts and pulse until only small chunks remain; season with salt and pepper.
Transfer broccolini and Halloumi to a platter. Dollop gremolata on top.
