1. For rosemary grilled octopus tentacle, heat oil or butter in a stockpot over medium heat and add onion. Stir to coat and lightly salt. Stir every 5−10 minutes until caramelized, about 1 hour.
2. Add the sachet of rosemary and waterand bring to a boil. Using tongs, dip octopus in the stock 3 times, then release into pot. Lower heat and simmer for 45 minutes uncovered, until tender. Remove from heat and wrap pot tightly with plastic wrap. Let cool.
3. Once cooled, remove octopus from liquid, then grill until browned, 3−5 minutes each side.Set aside.
4. For potato crema, heat stockpot over low heat and add oil or butter, swirling to coat pan. Add shallots, potato, thyme sachet, and salt and sweat 5−10 minutes, stirring frequently and adjusting heat to avoid browning. Potatoes should be knife-tender. Remove thyme sachet. (Note: At this point, Chef Compton smokes the potatoes using hickory chips for about 8 minutes. We have omitted this step for simplicity and because the dish has plenty of flavor otherwise.)
5. Place potatoes with chicken stock in a blender and purée. Add milk, emulsify with olive oil(adding enough oil to thicken), and pass through a chinois. Whip the remaining purée with crème fraiche and mayonnaise. Reserve at room temperature.
6. Gently warm the tomatoes, garlic, olives, capers, anchovies, and baby potatoes together over low heat. Meanwhile, broil the octopus about 1½ minutes.
7. To plate, spread potato crema down the center of each plate. Cut individual tentacles for each plate and cover with tomato caper mix. Add the oregano and parsley. Finish with sea salt and micro basil, if desired.