Heat the oil in a medium skillet until shimmering. Add the chicken and saute until lightly browned, about 5 minutes. Add onions an peppers and saute until vegetables are tender, about another 5 minutes.
Add tomato juice and taco seasoning; stir well. Simmer until very thick and chicken is well-coated with sauce, stirring often to prevent burning, about 7 minutes.
Preheat broiler and place a rack about 6" from heat.
Spread 2 tbsp salsa on each slice of bread. Divide the chicken mixture evenly amongst the slices of bread. Top each piece with 1/4 cup cheddar cheese. Place under broiler until cheese is melted and beginning to brown, about 5 minutes. Finish each sandwich with a few drops of lime juice before serving.