Make the aioli: Combine the sun-dried tomatoes, basil, garlic, lemon juice, and black pepper in a food processor and process until finely chopped. Add the mayonnaise and sour cream and pulse once or twice to blend. Transfer to a bowl and refrigerate until use.
Combine the oil and garlic powder. Rub the oil all over the steak and generously season with salt and black pepper.
Prepare the grill for direct cooking over medium-high heat (450° to 500°F). Preheat the Flat Top over direct medium-high heat for 10 minutes.
Prepare the Searwood Pellet grill for griddle setup with the warming rack in place. Preheat your Searwood Pellet Grill for 15 minutes on manual mode 6 with the full-size griddle insert or Weber Crafted Griddle insert in place.
Place the steak on the Flat Top and cook, with the lid closed, until browned on both sides, 6 to 8 minutes for medium-rare. Transfer to a cutting board and rest, indoors, for 5 minutes.
Place the steak on the griddle and cook, with the lid open, until browned on both sides, 6 to 8 minutes for medium-rare. Transfer to a cutting board and rest, indoors, for 5 minutes.
While the meat is resting, toast the bread, cut-sides down, on the griddle over direct medium-high heat until golden and well marked, about 1 minute.
While the meat is resting, toast the bread, cut-sides down, on the warming rack until golden and well marked, about 1 minute.
Thinly slice the steak against the grain. Smear the cut sides of the bread with some of the aioli. Arrange a layer of arugula over the bottom pieces. Place the steak on the arugula and top with additional arugula. Top the sandwiches with the bread. Serve with the remaining aioli for dipping.
