Bring a medium saucepan of water to a boil and gently drop in the garlic cloves. Boil gently for 10 minutes and then remove and drain the garlic cloves. Mash the garlic in a mortar and pestle or however else you choose.
Warm the olive oil in a small, heavy pot over medium-low heat. Add the anchovies and the garlic, bring to a simmer and lower the heat so the liquid is barely bubbling. Cook very gently for half an hour, stirring occasionally so that the garlic doesn't stick to the bottom of the pot.
Blanch and cut up the vegetables while the dip is simmering and arrange them on a large platter.
When the bagna cauda is ready, add lots of black pepper and some more salt if necessary. Serve warm with the vegetables.