In a food processor, combine olive oil, vinegar, and anchovies. Run processor until mixture emulsifies and gets thick. Taste and add more anchovies if desired.
Add in shallot and give it a couple good pulses. Set aside.
Get your grill nice and hot.
Cut cabbage in half, taking care to leave an even amount of the core in each half. Cut each half into four pieces, again, taking care to leave a piece of the core in each one so it does not fall apart.
Brush each piece of cabbage with a little olive oil on both sides.
Place cabbage pieces on grill and cover. Cook for about 5 minutes per side, until outer edges are nicely charred. Flip and repeat on other side. Transfer to platter.
Drizzle anchovy sauce over cabbage and serve immediately.