Prep the squid (10 minutes, ahead). Cut the bodies open lengthwise and lay them flat. Pat the bodies and tentacles very dry with paper towels, changing towels until they come away dry. Put them on a plate and season with the paprika and salt. Keep the squid in the refrigerator until the pan is ready.
Mix the mayonnaise. Stir the mayonnaise, garlic, and lemon juice together in a small bowl. Set it aside.
Char the corn (4 to 5 minutes). Heat a large skillet over high heat until a drop of water skitters off it. Add 1 tbsp (15 ml) of oil and the corn kernels in a single layer. Leave them alone for a minute, then stir. They are ready when they have brown spots and smell sweet and toasty. This is my skillet adaptation of the grilled corn used in published recipes. Toss with the onion, jalapeño, lime juice, and herbs, and move to a bowl.
Toast the rolls (2 minutes). Toast the split rolls cut side down in the same skillet, or under the broiler. Spread the mayonnaise on both halves.
Heat the pan for the squid. Wipe out the skillet, then heat it over high heat for a full 2 minutes. The pan should be hot enough that oil shimmers immediately.
Sear the first batch (2 to 3 minutes). Add 1 tbsp (15 ml) of oil, then half of the squid in a single layer. It should sizzle loudly the moment it lands. Cook the first side until deep golden, about 1 1/2 minutes, then flip and cook 30 to 60 seconds more, until the flesh is opaque. Move it to a plate. Wipe the pan, reheat for 30 seconds, and repeat with the rest.
Assemble right away. Cut the large pieces into bite-size strips. Layer tomato, arugula, squid, and a generous spoonful of corn salad onto each roll. Press the top on and serve while the squid is warm.