Preheat the oven to 400 degrees F. On a parchment lined baking sheet, toss the carrots with 1 tablespoon olive oil and season with salt. Roast in the oven until soft and caramelized, about 30 to 40 minutes. Transfer to a food processor and puree with the almonds, garlic, vinegar, paprika, chili, water, ¼ cup of olive oil, and ½ teaspoon salt until smooth. Taste for seasoning.
Pat the scallops dry and remove the muscle on the side. Season with salt.
Set a large heavy skillet over high heat. Add a thin coat of olive oil followed by the butter. When the butter is beginning to bubble and brown, carefully add the scallops in an even layer. Cook, without moving, for 1-2 minutes, until a dark crust has formed. Flip and cook for another 30 seconds. Remove to a serving platter browned-side up and serve alongside the romesco.