Remove and discard the bottom half of the broccolini stems. Cut the broccolini stems in half or quarters lengthwise, depending on the size of the stems. Don't cut the florets--just pull them apart. Place the broccolini on a sheet pan. Drizzle 3 tablespoons of olive oil on the broccolini and sprinkle with 1 teaspoon salt and 1/2 teaspoon pepper. Toss well, making sure the broccolini is lightly coated with oil. Spread the broccolini in one layer and roast for 10 to 15 minutes, tossing once with a metal spatula, until crisp-tender.
Meanwhile, slice the Cheddar 1/4-inch thick and break it into large crumbles. When the broccolini is ready, sprinkle the cheese on the broccolini and return to the oven for 3 to 4 minutes, until the cheese just melts. Squeeze on the lemon juice, taste for seasonings, and serve hot.