Preheat the oven to 300 degrees.
Roast the peppers by placing them on an open gas flame, turning them frequently with tongs until all the sides are charred black, 7 to 10 minutes. (Alternately, the peppers can be roasted under a broiler, or on top of a gas or charcoal grill.) Put the peppers in a plastic or paper bag, and cool for about 15 minutes.
Peel the peppers, remove the seeds and stems, and discard. Coarsely chop the peppers.
In a bowl, season the shrimp with the Essence, tossing to coat evenly. Heat 1 tablespoon of the oil in a large, non-stick skillet over medium-high. Add the garlic and cook, stirring, for 15 seconds. Add the shrimp and cook until the shrimp are just pink, 1 1/2 to 2 minutes. Remove from the heat and transfer to a plate to cool. Wipe the skillet clean.
In a bowl combine the 2 cheeses.
On a work surface, spread 3 of the tortillas. Sprinkle about 1/2 cup of the cheese over the bottom of each tortilla. Top each with 1/3 of the shrimp, mushrooms, and chiles. Sprinkle with 1/2 cup of the remaining cheese and top with a tortilla, pressing to adhere.
Add 1 teaspoon of the remaining oil to the pan over high heat. Add 1 assembled quesadilla and cook until the cheese starts to melt and the bottom starts to brown, about 3 minutes. Turn with a large spatula and cook until the bottom is golden brown, 2 to 3 minutes. Remove to a baking sheet and place in the oven to keep warm. Repeat with the remaining oil and quesadillas.
To serve, cut each quesadilla into 6 wedges and divide among plates. Serve with the poblano salsa on the side for dipping.
For the Smoked Mushrooms: Combine the mushrooms, oil, and Essence in a medium bowl and toss to coat evenly.
Prepare a stovetop smoker according to the manufacturer's instructions. Place the mushrooms on the grill pan and cover the smoker, leaving the lid ajar by 1-inch. Place over medium heat on top of the stove. When it begins to smoke, close the lid completely. Cook until the mushrooms are tender and completely smoked, about 20 minutes. Remove from the smoker and set aside until ready to assemble the quesadillas.
For the Adobo Chile Salsa: Combine all the ingredients in a blender and blend until just smooth but still slightly chunky. Transfer to a bowl and serve with the quesadillas.
This recipe yields 4 servings.