For the vinaigrette:
Combine the Anaheim chiles, poblanos, garlic, vinegar, lime juice, a few tablespoons of water, honey and salt and pepper, to taste, in a blender and blend until smooth.
With the motor running, slowly add the oil until emulsified. Add the parsley and blend 5 seconds longer. (There should be flecks of the parsley in the vinaigrette.)
For the quesadilla:
Preheat the grill to medium heat. Place 8 tortillas on a flat surface and evenly divide the Monterey Jack, cabrales, beef, goat cheese, bell pepper, Anaheim pepper and green onion among the tortillas (in that order). Season with salt and pepper, to taste, and sprinkle with Parmesan.
Stack the tortillas to make 4 (2-layer) tortillas and top each with the remaining tortillas. Combine ancho chili powder and canola oil, and brush the tops of the tortillas. Carefully place on the grill, ancho side down and grill until lightly golden brown, about 2 minutes. Flip the quesadillas over, close the lid of the grill and grill until the bottom of the tortillas are lightly golden brown and the cheese has melted. Remove from the grill to a cutting board. Cut into quarters, top each quarter with some of the vinaigrette and garnish with cilantro leaves. Transfer to a serving platter and serve.