Season the chicken with the allspice, turmeric, 1/2 teaspoon salt and a few grinds of pepper.
Heat 1 tablespoon olive oil in a large nonstick skillet or seasoned cast iron pan over medium-high heat. Add the chicken and cook until browned, 3 to 4 minutes per side. Transfer to a plate and set aside.
Add the apples, red onion and a pinch of salt to the skillet.
Cook the apple/onion mixture, stirring occasionally, until softened and lightly browned, about 4 minutes.
Add the lemon juice to the skillet and cook, scraping up any browned bits, 30 seconds.
Add the chicken broth and bring to a boil, then reduce the heat to maintain a simmer. Arrange the chicken on top of the apple mixture. Cover and cook until the chicken is cooked through and the sauce is slightly reduced, 6 to 9 minutes.