Preheat the oven to 475°F and line a medium baking dish with parchment paper. Coat the bottom with about a ¼-inch layer of salt, then place the whole carrots in the dish and sprinkle with a good amount of salt (see photo). Don’t worry, you won’t be eating all this salt in the final result. Roast the carrots until easily pierced with a fork, but not mushy. The timing will depend on the size and freshness of your carrots. Check them, starting around 40 minutes - my very large carrots took 60-90 minutes. This step can be done in advance.
Make the marinade: In a shallow dish or small bowl, combine the olive oil, rice vinegar, paprika, lemon juice, and several grinds of freshly ground black pepper.
Remove the carrots from the oven and let cool. Use your hands to rub off any excess salt. Use a knife to slice a thin strip off one side of the salty skin, and then use a peeler to peel the carrot into ribbons. If your peeler get's snagged on the soft carrot, that's ok, just slice pieces as thinly as you can with a sharp knife. Place the strips in the marinade and toss to coat. Transfer to the refrigerator and marinate for 15-30 minutes.
Serve with bagels, cream cheese, cucumber slices, capers, chives and/or dill.
If you have extra carrots, cover and refrigerate them in the marinade for up to 4 days.