Chicken and Green Chile Chilaquiles

1 3/4 pounds skinless, boneless chicken breast halves
2 1/2 cups chicken broth
1/2 cup dry white wine
1/2 lime, juiced
4 garlic cloves, 2 smashed and 2 minced, divided
salt
3/4 teaspoon dried Mexican oregano, divided
1 bay leaf
2 T olive oil
1 onion, chopped
2 to 4 jalapenos or serrano chiles, seeded and minced
1 (28-ounce) can whole tomatoes, crushed, with juices
3 poblano peppers, roasted, seeded and peeled, cut into thin strips
1/3 cup heavy cream
vegetable oil, for frying
12 corn tortillas, cut into 1/2-inch strips
sour cream, for serving
crumbled queso fresco, queso anejo, or Parmesan, for garnish

In a large saucepan combine the chicken, chicken broth, wine, lime juice, smashed garlic cloves, 1 teaspoon of salt, half of the oregano, and bay leaf and bring to a boil. Reduce heat to a simmer and cook for 10 to 12 minutes, or until chicken is just cooked through. Remove from the heat and let chicken cool in the broth, then remove and shred chicken into bite size pieces. Strain broth and reserve separately.

Heat the olive oil in a large saucepan and, when hot, add the onion and cook until soft and translucent, 4 to 6 minutes. Add the jalapenos and minced garlic and cook until fragrant, about 2 minutes. Add the tomatoes, remaining oregano, and 1 1/2 cups of the reserved chicken cooking liquid, and cook, stirring occasionally, until thickened and flavors have come together, about 30 minutes. Add the poblano strips and heavy cream and season, to taste, with salt. Cook until the sauce is thick enough to coat the back of a spoon, 5 to 10 minutes longer. Set aside.

Heat 2 inches of oil in a saucepan or deep-fryer and fry the tortilla strips, in batches, until light golden brown, 1 to 2 minutes. Drain on paper towels and season lightly with salt. Repeat with remaining tortilla strips.

Lightly grease an 8 by 11 1/2-inch casserole and lay half of the tortilla strips on the bottom of the casserole. Arrange half of the chicken over the tortilla strips, then spoon half of the tomato-poblano sauce over the chicken. Top with the remaining tortilla strips, remaining chicken and remaining tomato-poblano sauce, then sprinkle the top with the cheese and bake until bubbly and lightly browned on top, about 30 minutes. Serve while hot, with dollops of sour cream on top and garnished with some of the queso fresco.