Leaving root ends intact, quarter escarole lengthwise; brush with 1 tablespoon oil. Sprinkle chicken with 1/4 teaspoon each paprika and salt.
Take the escarole, chicken, corn, scallions and tomatoes out to the grill. Oil the grill rack (see Tips). Place escarole and scallions on the grill. Grill, turning occasionally, until charred in spots: 2 to 4 minutes for the scallions, 3 to 4 minutes for the escarole. Transfer to a plate. Place the chicken, corn and tomatoes on the grill, with the chicken on the hottest part and the tomatoes skin-side down. Grill the chicken, turning occasionally, until an instant-read thermometer inserted in the thickest part registers 165 °F, 8 to 16 minutes total. Grill the corn, turning occasionally, until tender, 8 to 12 minutes. Grill the tomatoes, turning once, until charred in spots, 6 to 8 minutes. Transfer to a plate.
Place the tomatoes in a blender; add garlic, vinegar, the remaining 2 tablespoons oil and the remaining 1/4 teaspoon each paprika and salt. Puree until smooth.
Cut off the root ends of the escarole; coarsely chop the leaves and put in a large bowl. Cut the corn kernels off the cobs and coarsely chop the scallions; add to the bowl. Toss the salad with 1/2 cup of the dressing. Divide among 4 plates. Thinly slice the chicken and divide among the salads. Drizzle with the remaining dressing and sprinkle with cheese and pine nuts.
Tips: Smoked paprika is made from smoke-dried red peppers and adds earthy, smoky flavor to food. Look for different types of paprika: sweet, bitter sweet and hot at well-stocked supermarkets, gourmet markets or online at tienda.com.
Toast chopped or sliced nuts in a small dry skillet over medium-low heat, stirring constantly, until fragrant and lightly browned, 2 to 4 minutes. Spread whole nuts on a baking sheet and bake at 350°F, stirring once, until fragrant, 7 to 9 minutes.
Oiling a grill rack before you grill foods helps ensure that the food won't stick. Oil a folded paper towel, hold it with tongs and rub it over the rack. (Do not use cooking spray on a hot grill.) When grilling delicate foods like tofu and fish, it is helpful to coat the food with cooking spray.