Bring a large pot of water to a boil. Season generously with salt. Cut off the top ½-inch of each artichoke and discard. Trim the stem and peel it using a vegetable peeler (the stem is delicious). If the leaves are sharp, snip off the tips using kitchen shears. Slip the artichokes into the boiling water. Cover, leaving a small crack for some steam to escape, and simmer until the base is tender when pierced with a paring knife, about 30 minutes. Transfer to serving plates. Do Ahead: The artichokes can be made up to 1 hour in advance, if you don’t mind serving them at room temperature (they taste just as good!).

Meanwhile, in a small saucepan, combine the butter, olive oil, garlic and anchovies. Cook over low heat, stirring occasionally and mashing the anchovies with the back of the spoon, until the anchovies are completely dissolved and the bagna cauda smells amazing, about 8-10 minutes. Remove from heat and stir in the lemon juice and Parmesan. Do Ahead: The bagna cauda can be made up to 1 hour in advance and left in the pan at room temperature.

To serve, reheat the bagna cauda gently (if needed) and divide it among 4 small ramekins or dishes. Serve the artichokes with the bagna cauda for dipping.