Preheat a grill to medium high, then prepare for indirect grilling (for a charcoal grill, bank the coals to one side of the grill; for a gas grill, turn off half the burners). Let the steaks stand at room temperature, 30 minutes.
Meanwhile, brush the grill grates with vegetable oil. Grill the tomato halves over direct heat until charred, 3 to 4 minutes per side. Let cool, then transfer to a mini food processor; pulse until slightly chunky. Add the butter, horseradish, tomato paste, paprika, garlic, 1 teaspoon salt and a few grinds of pepper. Process until smooth, scraping down the food processor occasionally. Transfer the tomato butter to a bowl and refrigerate until ready to use.
Season the steaks with salt and pepper. Place on the cooler side of the grill. Cover and cook until a thermometer inserted sideways into the center registers 110 degrees F, about 8 minutes per side. Move the steaks to direct heat and cook, uncovered, until charred and the thermometer registers 125 degrees F for rare to medium rare, about 2 more minutes per side. Transfer to a cutting board or baking sheet; spread the tomato butter on the steaks and let rest 10 minutes before slicing.