Cut broccoli into spears, leaving long stems attached to the flowerets. Peel the stems carefully, they break easily. Rinse the spears under cold running water. Leave them for 5 minutes to drain. Heat the oil over medium-high heat in a frying pan large enough to accommodate the broccoli in a single layer. When oil is hot, add garlic, and saute, turning and tossing until it turns golden, about 1 to 2 minutes. Add tumeric and immediately follow it with the broccoli. Spread the broccoli so that it lies in one layer. Let it sizzle undisturbed for 1 minute, then sprinkle on the salt. Turn the broccoli carefully with a flat spatula or a pair of tongs, and saute for an additional minute.
Reduce heat and cook, covered, until the broccoli is cooked but still crisp and dark green, about 8 to 10 minutes. Uncover, and continue cooking until all the moisture evaporates and the broccoli spears are glazed with garlic oil about 3 to 5 minutes. Check for salt and serve immediately.