Let the steak rest at room temperature for about 30 minutes before cooking. This helps it cook more evenly.
In a large skillet set over medium-low heat, add the olive oil and sliced garlic. Cook until the garlic is just golden and fragrant, 1 to 2 minutes.
Add the drained piquillos, salt, and sugar. Cook gently, flipping once or twice, until they’re heated through and slightly caramelized around the edges, about 10 minutes.
For a little brightness, add a splash of sherry vinegar just before serving. Reduce the heat to low and keep warm while you prepare the steak.
Pat the ribeye dry and season it generously all over with salt and black pepper, if using.
Heat a large cast-iron skillet over medium-high heat until very hot. Add a drizzle of oil, then carefully place the steak in the skillet. You should hear a big sizzle!
Sear for 3 to 4 minutes, undisturbed, until it develops a deep golden crust. Flip and sear the other side for another 3 to 4 minutes.
Reduce the heat to medium-low and add the butter and fresh thyme to the pan. As the butter melts and foams, tilt the pan slightly and spoon the melted butter over the steak for 1 to 2 minutes to baste it and infuse it with the thyme flavor. Continue to cook until a meat thermometer reaches 130°F for medium-rare.
Transfer the steak to a cutting board and let it rest for 10 minutes before slicing. This is important for juicy steak!
When ready to serve, slice it against the grain and arrange it on a platter, scattering the warm Basque-style piquillo peppers around it. Spoon some of the garlicky piquillo pepper oil over the top and finish with a sprinkle of flaky salt and freshly chopped parsley, if desired.