Step 1Heat the oven to 400 degrees.
Step 2Scrub the potatoes and toss them with 1 tablespoon of the oil, the garlic, thyme sprigs and bay leaf in a roasting pan. Cover the pan with foil and roast until the potatoes are tender, 50 minutes.
Step 3Cut away the broccoli stems where they start to get woody and discard. If the heads are large, cut them into florets. Cook the broccoli in boiling salted water until crisp-tender, about 3 minutes. Drain the broccoli, cool, then cut the spears into 2 or 3 pieces.
Step 4Mix the capers, olives and 1 tablespoon olive oil together in a small bowl.
Step 5Crumble the cooled potatoes with your hands into 5 or 6 pieces each. Squeeze the roasted garlic out of the skins and set aside.
Step 6Heat a large, heavy skillet, preferably cast iron, over medium heat. Add 3 tablespoons of oil, let it get hot, then add the potatoes. Cook the potatoes until they're golden brown and crisp in spots, stirring occasionally, about 10 minutes.
Step 7Add 1 tablespoon of the oil to the skillet and then add the shallots, broccoli, roasted garlic and thyme leaves and toss together with the potatoes. Cook the hash until the broccoli is hot, the shallots are translucent and the potatoes have begun to break down a bit, 3 to 4 minutes. Stir in the caper-olive mixture, and taste for seasoning, adding salt as needed.