Heat the oil in a small skillet over medium. Add the garlic and chile, and cook, stirring frequently, until the garlic is soft and lightly browned, usually 1 to 2 minutes. Pour oil, garlic and chile into a blender jar or food processor. Add the lime juice, cilantro and 1 scant teaspoon salt and ¼ teaspoon black pepper. Process until smooth.
Pour 1/3 of the garlic mixture over the chicken breasts, spreading it evenly over all sides.
Heat a grill pan or gas grill over medium to medium-high (or start a charcoal fire and let it burn until the coals are medium hot and covered with white ash). Lightly brush or spray the onion slices with oil; sprinkle with salt. Lay the chicken and onion on to the grill pan or grill. Cook until the chicken is just cooked through and the onion is well browned, 3 to 4 minutes on each side. Chop the onion into small pieces and scoop into a bowl.
Pit and peel the avocados, scooping the flesh in with the onion. Add another 1/3 of the garlic mixture, then coarsely mash everything together with an old-fashioned potato masher, large fork or back of spoon. Taste and season with salt, usually about ½ teaspoon.
Scoop the sliced romaine into a large bowl. Drizzle on the remaining 1/3 of the garlic mixture and toss to combine. Divide between 4 dinner plates. Scoop a portion of the guacamole into the center of each plate. Cut each breast into cubes and arrange over the guacamole. Sprinkle each plate with queso añejo (or its substitute) and you’re ready to serve.