In a large skillet, heat the olive oil over medium heat. Add the diced onion and sauté for about 5 minutes, or until the onion becomes translucent. Next, add the minced garlic and sauté for an additional minute until fragrant.
Gradually add the fresh spinach leaves to the skillet, stirring as you go. The spinach will wilt down significantly; keep stirring until all the spinach is wilted and tender, which should take about 3-4 minutes.
Remove the skillet from heat and stir in the fresh lime juice, chopped cilantro, green chiles, ground cumin, and chili powder. Season with salt and black pepper to taste. Then, fold in the sour cream (or Greek yogurt) until everything is well combined.
Preheat your oven to 350°F (175°C). Spread a thin layer of green chile enchilada sauce on the bottom of the baking dish. Take a tortilla, spoon about 1/3 cup of the creamy spinach mixture onto the center, and sprinkle with a little Monterey Jack cheese. Roll the tortilla tightly, placing it seam-side down in the baking dish. Repeat with the remaining tortillas.
Once all the enchiladas are in the dish, pour the remaining green chile enchilada sauce over the top. Sprinkle the remaining Monterey Jack and Cheddar cheese evenly over the sauce.
Cover the baking dish with aluminum foil and bake in the preheated oven for 20 minutes. Then, remove the foil and bake for an additional 10-15 minutes, or until the cheese is bubbly and slightly golden.