Preheat oven to 350°. In a large skillet over medium-high heat, heat lard and oil. Add poblano chiles, onion, garlic, and 2 teaspoons salt. Cook, stirring often, until vegetables are softened and beginning to brown, 8 to 10 minutes.
Add green chiles and stock, scraping up bits from bottom of skillet. Bring to a simmer and cook, stirring occasionally, until sauce has slightly reduced, 4 to 6 minutes. Transfer mixture to a blender; reserve skillet. Add cilantro to blender and blend until smooth, 30 seconds to 1 minute.
Place a tortilla directly on gas burner over high heat. Toast, turning once, until edges are slightly charred, about 30 seconds. Transfer to a cutting board and repeat with remaining tortillas, stacking them on top of each other. Cut tortilla stack in half.
Pour one-third of sauce into bottom of reserved skillet. Top with half of tortillas in an even layer, followed by 2 cups chicken, 3/4 cup beans, and 3/4 cup cheese. Pour in another one-third of sauce. Top with layers of remaining tortillas, 2 cups chicken, and 3/4 cup beans. Pour over remaining one-third sauce and top with remaining cheese.
Bake enchiladas until warmed through and cheese is melted, 18 to 22 minutes.
Dollop with sour cream. Top with cilantro, avocado, and radish.