Cut a square of tinfoil. Slice off the top of the head of garlic to expose the cloves inside. Set the head of garlic in the centre of the tinfoil, cut side up. Drizzle olive oil onto exposed cloves. Sprinkle with salt and pepper. Close the tinfoil over the garlic to create a package. Set on middle rack of the oven and cook for 30-35 minutes.
Remove garlic and let cool slightly. Carefully unwrap foil and remove cloves by squeezing roasted flesh out of bulb. Place one cup (240 milliliters) olive oil, butter, anchovies, and garlic flesh in a saucepan and heat over low, stirring often, for 15-20 minutes.
Meanwhile, toss fennel, carrots, endives, red pepper, zucchini, and thyme with 3 tablespoons (45 milliliters) olive oil and salt and pepper. Spread out on a parchment paper-lined baking sheet. Bake at 425 F (230 C) for eight to ten minutes or until just fork-tender.
Remove garlic mixture from heat and transfer to a food processor or blender and blitz until smooth. Pour into a serving dish and serve with prepared accompaniments.
Optional: Serve bagna cauda in a fondue dish heated by a candle.