Heat the oil in a small skillet over medium. Add the garlic and chiles and cook, stirring frequently, until the garlic is soft and lightly browned, usually 1 to 2 minutes. Pour the oil, garlic and chile into a blender jar or food processor. Add the lime juice, cilantro, black pepper and 1 scant teaspoon salt. Process until smooth.
Place the chicken breasts in a bowl and pour one-third of the garlic dressing over them, spreading it evenly over all sides.
Heat half the burners of a gas grill over medium-high (or start a charcoal fire and let it burn until the coals are covered with white ash but still quite hot, then bank the coals to one side). Lightly brush or spray the onion slices with oil; sprinkle with salt. Lay the chicken and onion directly over the heat of the grill. Sear the chicken on both sides, then move it away from the direct heat to finish cooking, usually about 5 minutes. Cook the onion until is it well browned, 3 to 4 minutes on each side. Chop the onion into small pieces and scoop into a bowl. Set chicken aside to cool. (All of the grilling can be done in a grill pan on a stove over medium heat.)
Pit the avocados and scoop the flesh in with the onion. Add another one-third of the garlic dressing, the coarsely mash everything together with an old-fashioned potato masher, large fork or the back of a big spoon. Taste and season with salt, usually about 1/2 teaspoon.
Scoop the sliced romaine into a large bowl. Drizzle on the remaining garlic dressing and toss to combine. Divide among the four dinner plates.
Scoop a portion of the guacamole into the center of each plate. Cut each chicken breast into cubes and arrange over the guacamole. Sprinkle each plate with queso añejo (or its substitute) and you're ready to serve.