Prepare grill for medium-high heat. Combine chile and 2 Tbsp. vinegar in a small bowl; set aside.
Place onions on a rimmed baking sheet and drizzle with 2 Tbsp. oil; season with salt and pepper. Turn to coat. Grill onions directly on grate until lightly charred and softened, about 2 minutes per side. Transfer to a large bowl and toss with remaining 2 Tbsp. vinegar; let cool.
Coarsely chop ½ cup grilled onion and return to bowl. Add chile and soaking liquid, cucumber, oregano, and 2 Tbsp. oil and toss to combine; season with salt and pepper. Serve drizzled with more oil.