In a 2- to 3-quart pan or an electric skillet over medium heat, stir olive oil, butter, anchovies, and garlic, breaking up anchovies and garlic slightly with the back of a fork, until butter is melted and mixture is bubbling, about 5 minutes. Remove from heat and let stand at least 10 minutes or up to 30 minutes.
Meanwhile, arrange the radicchio, endive, peppers, green onions, and mushrooms on a serving platter. Reheat butter mixture gently over low heat, then pour into a bowl or ramekin. Alternatively, pour into a 1 1/2- to 2-cup ceramic fondue pan or metal chafing dish in a water-bath jacket. Set over a votive candle or an ignited alcohol or canned solid-fuel flame. Adjust heat to lowest setting under fondue pan, to medium under chafing dish. Pass platter of vegetables and bread for guests to dip in warm bagna cauda.