Bring a large pot of salted water to a rolling boil. Prepare a large bowl of ice water and set it aside. Cook the broccolini in the boiling water until crisp tender and very bright green, 3 to 4 minutes. Transfer the broccolini to the ice water to stop the cooking, then drain very well.
In a large skillet, heat the butter and oil over medium-high heat. Add the broccolini and sauté until lightly charred, 5 to 6 minutes.
Add the garlic, lemon juice, and 1 teaspoon salt. Sauté, stirring constantly so the garlic doesn’t burn, until the broccolini is charred and the garlic is just golden, about 2 minutes more.
Remove the broccolini and garlic to a platter. Add the lemons cut-sides down and let char, 2 to 3 minutes, if desired. Serve with the broccolini.