Bring a large pot of water to a boil.
Cut fatter broccolini stems lengthwise then cut thin florets and stalks in half across. Salt boiling water and add broccolini; cook 2-3 minutes to tender-crisp then remove with a spider to cool—keep water at a low boil for the pasta. Chop in half or thirds.
Heat oil, 2 turns of the pan, in a large skillet over medium-heat. Add broccolini to hot olive oil, season with salt and pepper, and toss with garlic. Add cream and nutmeg, and simmer to reduce over medium-high heat.
Turn the heat up under the pasta pot then cook pasta to al dente.
Turn off heat on broccolini once cream is reduced by half and stir in 1 cup cheese, a few fat handfuls. Toss linguine with cream, broccolini and remaining cheese. Sprinkle breadcrumbs over pasta to serve.