Heat oil in a medium-size high-sided skillet over medium-high. Sprinkle chicken evenly with salt. Place chicken in skillet, skin side down; cook, undisturbed, until skin is golden brown and crispy, about 8 minutes. Transfer chicken to a plate, reserving rendered fat in skillet; set aside.
Add mushrooms to skillet; cook over medium-high, stirring occasionally, until golden brown, 6 to 8 minutes. Add shallot; cook, stirring often, until softened, about 3 minutes. Add garlic, curry powder, and turmeric; cook, stirring constantly, until fragrant, about 30 seconds.
Stir in coconut milk, 1/3 cup water, and fish sauce until combined. Return chicken thighs to skillet, skin side up. Bring to a simmer over medium-high. Reduce heat to medium-low; cook, uncovered, swirling pan occasionally, until sauce is thickened and a thermometer inserted in thickest portion of chicken thigh registers 165°F, 20 to 25 minutes.
Transfer chicken thighs to a shallow serving bowl. Add evaporated coconut milk to skillet, and stir to combine; pour curry evenly over chicken thighs. Garnish with cilantro and chile oil. Serve with rice.
