Fry tortilla strips:

Heat oil in a 3 1/2-quart flameproof nonreactive shallow casserole or deep skillet to 375°F. Fry tortillas in batches, turning occasionally, until golden, about 1 minute, and transfer to paper towels to drain (return oil to 375°F between batches). Carefully pour off all but 1 tablespoon hot oil into a large metal bowl to cool before discarding.

Make chilaquiles:

Preheat oven to 375°F. Cook onion in oil in casserole over moderately high heat, stirring, until softened. Add broth and turkey and simmer, uncovered, stirring, until liquid is reduced to about 1/2 cup, about 15 minutes. Add 2 1/2 cups salsa and bring to a boil. Remove from heat and toss with Monterey Jack and tortilla strips. Bake chilaquiles, uncovered, in middle of oven until bubbling, about 15 minutes. Whisk together crème fraîche and milk. Serve chilaquiles topped with crème fraîche, cilantro, and cheese.