Put the spinach in a large microwave-safe bowl; cover with plastic wrap and pierce a few times to vent. Microwave until it starts wilting, about 4 minutes. Uncover and stir, then microwave until completely wilted, about 2 more minutes. Transfer to a colander and rinse under cold water; squeeze dry and set aside.
Heat the olive oil and butter in a large saucepan over medium-high heat. Add the onion, poblanos and 1/2 teaspoon each salt and pepper and cook, stirring, until the onions are translucent, about 4 minutes. Add the mushrooms and cook, stirring, until tender, about 5 minutes. Reduce the heat to medium; add the garlic and flour and stir until incorporated, about 2 minutes.
Gradually stir in the chicken broth and bring to a simmer. Add the sour cream and cook, stirring, until the liquid thickens, about 5 minutes. Remove from the heat and stir in all but a few tablespoons of the gruyere, then stir in the muenster and cilantro. Transfer to a food processor and add the spinach; pulse until finely chopped.
Preheat the oven to 400 degrees F. Spread the dip in a 1-quart baking dish and sprinkle with the reserved gruyere. Transfer to the oven and bake until the cheese melts, about 6 minutes. Serve with bread.