Quick game plan. Simmer the octopus until tender, boil the potatoes, slice everything, then go heavy on olive oil, paprika, and flaky salt.
Prep note: what “cleaned” means. Ideally, ask your fishmonger to clean it. At home, “cleaned” means the beak is removed (it is where the tentacles meet underneath) and the eyes are cut out. Many store-bought octopus are already cleaned, especially frozen ones.
Simmer the octopus (tenderness over the clock). Fill a large pot with water and bring it to a gentle simmer. I usually do not salt the water and season at the end, but you can add a pinch if you want. If using bay, garlic, and lemon, add them now for a subtle background flavor.
Optional traditional step (for the purists): “scare” the octopus. Using tongs, dip the tentacles into the simmering water for 2 to 3 quick dips, then fully submerge. This helps the tentacles curl and can help keep the skin intact.
Keep the pot at a low, steady simmer for 60 to 120 minutes, depending on size and thickness, until a knife slides into the thickest part easily. If foam rises, skim it. Keep the heat calm. Octopus likes a spa day, not a mosh pit.
Cook the potatoes. While the octopus simmers, place potatoes in a pot, cover with cold water, and add a good pinch of salt. Bring to a simmer and cook until tender, 20 to 30 minutes. Drain, cool just enough to handle, then slice into 1/2-inch rounds.
Skin-on or peeled? Yukon Gold skins are thin and totally fine here, so I usually leave them on. If you want a cleaner, classic look, peel after boiling while they are still warm.
Optional traditional move: Once the octopus is done, you can boil the potatoes in the octopus cooking liquid instead of plain water for extra flavor and that classic rosy tint. If you do this, fish out any aromatics first, keep the liquid at a gentle simmer, and cook the potatoes as usual.
Cool, then slice. When the octopus is tender, transfer it to a cutting board and let it cool 10 minutes. Slice tentacles into 1/2-inch coins. (Cooling helps it stay juicy and makes slicing cleaner.)

Optional crisp-edge step. Heat a large skillet over medium-high heat with 1 to 2 tablespoons olive oil. Sear octopus slices for 60 to 90 seconds per side until lightly browned at the edges.
Assemble. Arrange potato rounds on a platter. Pile octopus on top. Drizzle generously with olive oil. Dust with smoked paprika (and hot smoked paprika if using). Finish with flaky salt, a squeeze of lemon, and parsley if you want it.

Taste and adjust. Take one bite and decide what it needs. Usually it is more flaky salt, more lemon, or more olive oil. Sometimes it is all three. Trust your tongue.