Preheat the broiler and position an oven rack about 6 inches from the heat.
Heat the olive oil in a large skillet over medium-high heat.
Add the chopped red onion and garlic and cook, stirring, until softened, about 2 minutes.
Stir in the lime juice and honey, then add the salsa verde and warm the mixture until heated through.
Remove the skillet from heat and stir in the sour cream until the sauce is smooth and combined.
Place the shredded chicken in a mixing bowl and add the sauce gradually, stirring to coat; you may not need all the sauce depending on how moist your chicken is.
Pile the sauced chicken onto the bottom halves of the split rolls, then sprinkle evenly with the shredded Monterey Jack cheese.
Place the assembled sandwich bottoms on a baking sheet and broil just until the cheese melts and bubbles, watching closely to avoid burning.
Set the roll tops in place and serve immediately.