Trim off any hard ends from the broccoli stalks as well as any very coarse leaves. Bring a large quantity of salted water to the boil in a very large saucepan and plunge in the broccoli. With the lid on, quickly bring the water back to the boil, then remove the lid and briskly simmer the broccoli for about 2 minutes. Drain thoroughly.
Meanwhile, heat both oils in the pan over a medium heat. Fry the garlic for 2-3 minutes, stirring constantly, until the slices are light golden. Throw in the broccoli and fry it for 2-3 minutes, tossing frequently, until the stalks are piping hot and imbued with the garlicky oil.
Serve immediately with the sesame seeds sprinkled over the top.