When you push broccoli beyond that disappointing just-too-done state (and throw in a whole lot of olive oil, bubbling lazily with garlic, anchovy, and hot peppers) you find yourself with a miraculous substance -- essentially broccoli confit. The florets trap all the oil's richness, and the stems melt away.
It would be fitting spooned up onto some sturdy bread, blanketing a good ridge-y pasta, layered onto a pizza, or anointing a sandwich. Finamore says it also works with cauliflower.
While the water is heating, cut the florets off the broccoli. Peel the stems and cut them into rather thick slices, about 1/3 inch.
When the water comes to a boil, add the broccoli and cover the pot to bring it back to a boil quickly. Blanch the broccoli for five minutes. Drain.
Put olive oil and garlic in a large skillet over medium heat. When the garlic starts to sizzle, add the hot peppers and anchovies. Cook, giving a stir or two, until the anchovies melt. Add the broccoli, season with salt and pepper, and stir well. Cover the skillet, turn the heat to very low, and cook for two hours. Use a spatula to turn the broccoli over in the skillet a few times, but try not to break it up. It will be very tender when done.
Use a slotted spoon to transfer the broccoli to a serving dish. It is delicious hot or at room temperature.