Preheat oven to 375°F. Prepare a 9x13-inch casserole dish with a coating of butter or nonstick cooking spray. Set aside.
Pat the shrimp dry with paper towels. Roughly chop them into bite-sized pieces and set aside.
In a large skillet over medium heat, melt the butter. Add the garlic, onion, and jalapeño pepper. Sauté 1-2 minutes, until the vegetables become fragrant and begin to soften.
Add the shrimp to the skillet and sprinkle with a pinch of salt and pepper. Sauté just until the shrimp start to turn pink, about 2-3 minutes. They should still be slightly undercooked, as they will finish cooking in the oven. Remove the pan from heat.
Using a slotted spoon, transfer the shrimp and vegetable mixture to a medium bowl, leaving behind any excess liquid in the pan. This prevents the enchiladas from becoming soggy while baking.
In a separate bowl, whisk together the salsa verde, chicken broth, and sour cream to make the cheese sauce. Add 1/4 cup of shredded Monterey Jack cheese and stir until combined.
Stir about 2–3 tablespoons of the cheese sauce into the shrimp mixture, just enough to bind the filling. Reserve the remaining cheese sauce.
Spoon roughly 2 tablespoons of the shrimp filling into the center of each tortilla. Roll up each tortilla tightly and place seam side down in the prepared casserole dish.
Pour the reserved cheese sauce evenly over the enchiladas. Sprinkle the remaining 1/2 cup of Monterey Jack cheese over the top.
Bake uncovered for 20-25 minutes, until the cheese is melted and bubbly and the dish is heated through.
Remove from oven and let rest 5-10 minutes before serving.