FOR THE BEURRE MONTÉ: Bring water to simmer in small saucepan over medium-high heat; reduce heat to maintain very gentle simmer. Add 1 piece butter and cook, whisking constantly, until butter is melted, 20 to 30 seconds. Continue to cook, whisking in butter 1 piece at a time until all butter is incorporated and sauce has consistency of thin gravy, 3 to 4 minutes. Whisk in gochujang, lime juice, and sugar, and season with salt to taste. Reduce heat to lowest possible setting and cover to keep warm.

FOR THE PORK: Adjust oven rack to middle position and heat oven to 275 degrees. Set wire rack in rimmed baking sheet and lightly spray rack with vegetable oil spray.

Pound each tenderloin to 1-inch thickness. Halve each tenderloin crosswise. Sprinkle each steak with ½ teaspoon salt and ⅛ teaspoon pepper. Place steaks on prepared wire rack and cook until meat registers between 137 and 140 degrees, 25 to 35 minutes.

Move steaks to 1 side of rack. Line cleared side with double layer of paper towels. Transfer steaks to paper towels, cover with another double layer of paper towels, and let stand for 10 minutes.

Pat steaks until surfaces are very dry. Heat oil in 12-inch skillet over medium-high heat until just smoking. Increase heat to high; place steaks in skillet; and sear until well browned on both sides, 1 to 2 minutes per side. Transfer to carving board and let stand for 5 minutes.

Slice steaks against grain ¾ inch thick and transfer to serving platter. Season with salt to taste. Whisk sauce vigorously to blend in any butterfat that has collected on surface. Spoon sauce over pork and serve.