Put the cucumber slices in a colander and sprinkle with salt and pepper; toss. Let sit 20 minutes to drain.
Meanwhile, heat 1/4 cup olive oil in a medium saucepan over medium heat. Add the onion, coriander and turmeric and cook, stirring occasionally, until softened, about 10 minutes. Add 1 cup water, the cilantro stems and carrots and bring to a simmer. Cover and cook, stirring occasionally, until the carrots are tender, about 10 minutes. Transfer the vegetable mixture to a food processor and puree until smooth, at least 1 minute. Season with 1 teaspoon salt and a few grinds of pepper; cover to keep warm.
Gently squeeze out any excess water from the cucumbers and transfer to a medium bowl. Stir in 2 tablespoons olive oil, the vinegar, cilantro leaves, garlic and jalapeno; season with pepper.
Heat a large cast-iron skillet over high heat. Season the lamb chops with salt and pepper; rub all over with the remaining 2 tablespoons olive oil. Cook the lamb chops two at a time, turning once, until browned, 6 to 7 minutes for medium doneness. Serve with the carrot puree and cucumber salad.