sometimes labeled pimentin picante or pimenton de la vera picante; available at specialty foods stores and from tienda.com.
Whisk first 4 ingredients in small bowl. Gradually whisk in 2 tablespoons olive oil. Season with salt and pepper. DO AHEAD Aioli can be made 2 days ahead. Cover prepared aioli and chill.
Heat vegetable oil in heavy large skillet over medium-high heat. Add onions, sprinkle with salt, and cook until beginning to brown, stirring often, about 10 minutes. Reduce heat to medium and continue to cook until onions are soft and deep golden brown, stirring often, about 20 minutes longer. Season with salt and pepper. DO AHEAD: Caramelized onions can be made 2 hours ahead. Let stand at room temperature.
Prepare barbecue (medium-high heat). Place chicken breasts between 2 sheets of plastic wrap. Pound with mallet to 1/3-inch thickness. Sprinkle with salt and pepper. Grill chicken until cooked through, about 3 minutes per side. Transfer to work surface. Brush cut side of rolls with extra-virgin olive oil. Grill rolls, cut side down, until grill marks appear, about 1 minute. Spread rolls with some of aioli. Assemble sandwiches, dividing chicken, caramelized onions, and watercress among rolls. Serve warm.