Preheat the oven to 400F and line a baking sheet with parchment paper.

Place chicken on one side of the baking sheet and poblano and bell peppers on the other side.

Roast, stirring peppers once or twice until peppers are browned and chicken is just cooked through, 40 to 45 minutes.

Cool slightly, then remove and discard skin and bone from chicken and shred meat.

Dice peppers.

Meanwhile, place cashews in a small bowl and cover by about 1 inch with boiling water.

Let soak 30 minutes.

Drain and discard soaking liquid.

In a blender, combine cashews, 1/2 cup water, vinegar, salt and cayenne, and blend until smooth; add more water a tablespoon at a time, if necessary, to make a smooth puree.

Reduce the oven temperature to 375F.

Layer 1/3 of the tortillas in the bottom of a 9x13-inch casserole dish.

Combine salsa and broth, and spoon 1/2 cup of the mixture over the tortillas.

Top with half the chicken and half the roasted peppers, and drizzle with half of the cashew cream.

Repeat layering again, finishing with a layer of tortillas and salsa mixture.

Sprinkle with cheese and bake until bubbling and golden brown, about 35 minutes.

Let cool 10 minutes before cutting and serving.