In a large skillet, heat the oil over medium heat. Add the bell peppers, onions, and garlic. Cook, stirring, until the veggies are softening, 3 to 4 minutes.
Add the turkey, ginger, salt, and pepper and continue to cook, breaking up the turkey into very small pieces, until the turkey is cooked through and no longer pink, 4 to 6 minutes. Drain off excess fat (sometimes not necessary with turkey!)
Add the curry powder, garam masala, cumin, and cayenne (if using). Cook, tossing, toasting the spices until fragrant, about 2 minutes.
Stir in the tomato paste until well combined. Add the crushed tomatoes, beef broth, and coconut sugar. Stir until well combined and let come to a light simmer. Cook, simmering and stirring often, for the flavors to meld and the sauce to thicken a bit, about 4 minutes.
Reduce heat to low and stir in the Greek yogurt until just combined. Taste and add more salt, as desired.
Garnish with chopped cilantro. To serve, ladle the curry sloppy joe mixture onto the halved potato buns. Top with thinly sliced serrano peppers.