In a skillet, heat the oil until shimmering. Season the chicken livers with salt and pepper, add them to the skillet and cook over high heat, turning once, until crisp and browned but still slightly pink in the center, about 5 minutes. Using a slotted spoon, transfer the livers to a plate.
Add the shallots to the skillet and cook over high heat until softened, about 1 minute. Add the port and cook, scraping up any browned bits, until reduced to a thick syrup, about 2 minutes. Add the cream and boil until thickened, about 2 minutes. Return the livers to the skillet and toss to coat with the cream. Transfer to a bowl and coarsely chop the livers with a fork. Add the chives and season with salt and pepper. Refrigerate until slightly chilled before serving on toasts.
The chicken liver pâté can be refrigerated for up to 3 days.