Combine the anchovies and garlic in a mortar and pestle, and crush them into a paste (or if you don’t have a mortar and pestle, mince them very small).
Melt the butter in a cast iron skillet over medium high heat, add the anchovy/garlic paste, and cook, stirring occasionally, until paste starts to brown and crisp up, about 2-3 minutes. Add 3 Tbsp olive oil, and cook 1 minute more. Transfer everything to a heatproof container (such as a mason jar or measuring cup), and set aside.
Keeping the cast iron pan over medium high heat, add the remaining 1 Tbsp olive oil. Once hot, add the torn bread, and cook tossing occasionally, until bread is golden and crispy, about 3-4 minutes. Remove bread from pan and set aside.
Return pan to medium high heat. Place the radicchio quarters cut sides down in the hot pan, searing until sides are just starting to char, about 1 minute per side. Turn each quarter once to char both cut sides. Depending on how big your pan is, you may need to sear the radicchio in batches. Once seared, transfer radicchio to a plate (if the radicchio has a tough core, cut it off the bottoms of the quarters before arranging on the plate).
Whisk 1 Tbsp balsamic vinegar into the warm anchovy/olive oil mixture. Pour evenly over the seared radicchio. Top evenly with the torn bread, and sprinkle arugula and shaved Parmesan over the top. Serve immediately.