In a medium saucepan, combine carrot juice, 1/4 cup wine, lemon juice, shallot, and salt. Bring to a boil over high heat. Reduce heat just enough to maintain a steady boil and cook until slightly thickened and reduced to about 1/2 cup. It should be the consistency of pulpy orange juice, 20 to 30 minutes.
Reduce heat and very gently simmer carrot reduction. Whisk in butter 1 piece at a time, waiting until each piece is melted and incorporated before adding another. The sauce should be thickened enough to lightly coat the back of a spoon. Return the sauce to a simmer, whisking constantly. Drop in and submerge the bundle of marjoram and remove from the heat. Let marjoram infuse sauce while cooking the scallops.
Heat oven to 175 degrees F. Turn off the oven and put in a platter to warm. If the scallops have a small, loosely attached, rectangular piece of white muscle on their sides, pull it off and discard it. Pat the scallops dry with paper towels and season all sides lightly with salt and pepper. Heat the oil in a large heavy-bottomed saute pan over high heat until it is very hot and smoking. Using tongs, carefully place the scallops flat side down in the pan. Let the scallops cook undisturbed until they are golden brown, 2 to 3 minutes. Turn and cook until the other side is browned, 1 to 2 minutes. Transfer scallops to warmed platter. Reduce heat to medium and pour in remaining 1/4 cup wine. Scrape up browned bits, then add liquid to carrot mixture.
Remove marjoram bundle from carrot sauce, squeeze it dry, and discard. Reheat sauce over medium heat, whisking constantly. Taste and adjust seasoning. For a very smooth texture, pass sauce through a very fine strainer. If there is a problem with sauce separating or butter coming to surface in droplets, blend it with an immersion blender for a few seconds. Divide scallops among warmed dinner plates and pour carrot sauce around them.
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