Prepare a grill for medium heat. Finely grate zest from lemon half into a small bowl, then squeeze in juice. Finely grate whole garlic clove into bowl and mix in mayonnaise. Season garlic mayo with salt and set aside.
Place a large cast-iron skillet on grill and heat 1/4 cup oil in skillet. Add sliced garlic, shallots, and chile and cook, stirring often, until just softened, about 2 minutes. Add tomato paste and cook, stirring often, until paste darkens slightly, about 1 minute. Add tomatoes and a pinch of salt and cook, stirring occasionally, until tomatoes soften and release their juices, about 4 minutes. Add wine and cook until it is almost reduced by half and no longer smells boozy, about 3 minutes.
Add clams and butter to skillet and cover (if you don’t have a lid that fits, use a sheet of foil). Cook until clams have opened, 6–10 minutes, depending on size of clams and heat level. Remove skillet from grill; discard any clams that don’t open. Sprinkle with chives.
Meanwhile, drizzle bread with remaining 3 Tbsp. oil and season lightly with salt. Grill until golden brown and crisp, about 3 minutes per side.
Serve clams with toasted bread and reserved garlic mayo.