Break down the broccoli by first separating the trunk, the branches, and the florets. Lop off the florets by positioning your knife right up next to the leaves so that the stem side ends up with as many branches as possible. Then, cut the trunk away from the branches.
Peel the trunk with a sharp knife or a spoon, discarding the tough exterior, and slice it up into coins. Slice the stems from each part of the broccoli into 1/4-inch cross-sections. The shapes of the pieces will be irregular like splashes of spilled milk, but they should be the same thickness as the coins of the trunk. Finally, slice away any stem that's still attached to the florets; you want the florets to be so small that they barely hold together at all, just clusters of leaves rather than tiny trees. Put the leaves in one bowl and all stem slices in another.
In a very large cast-iron skillet, heat the oil, anchovies (straight out of the jar), and garlic over medium heat until the anchovies break up and the oil becomes fragrant, about 5 minutes. Remove the garlic, turn the heat up a little, and add your stem slices in a single layer. They should have a little wiggle room to breathe; if yours are crowded, fry in batches with another batch of flavored oil. Season with a pinch of salt.
Let the stems get dark brown on one side, about 10 minutes, and then flip to sear the other side. When well-caramelized, stems should be honey-colored all over with just a trace of light green.
Turn down heat to medium low, stir, and pile the floret pieces on top of the stems. Add a splash of white wine or beer (whatever you're drinking while you cook), cover, and let steam for 1 minute, until florets are a vivid bright green. Using a slotted spoon, transfer all broccoli to a large bowl, toss with lemon juice, and season with salt and pepper. Serve alongside meats or fish, or pile on a piece of well-buttered toast.