Line a 6-inch square pan or a 9-by-5- inch loaf pan with plastic wrap; set aside. Melt 1/4 cup butter in a large skillet over medium-high heat. Add livers, mushrooms, shallots, garlic, and cayenne pepper. Cook, stirring occasionally, until livers are cooked through, about 5 minutes. Stir in 1 teaspoon salt, the wine, and thyme. Reduce heat to low. Cover; cook until liquid has reduced and mushrooms are very soft, about 10 minutes. Let cool.
Transfer mixture to a food processor. Add remaining 3/4 cup butter and 11/2 teaspoons salt; process until smooth, about 3 minutes. Transfer to prepared pan, and refrigerate until cool, about 30 minutes.
Sprinkle gelatin over 3 tablespoons pomegranate juice in a bowl; let stand until softened, about 5 minutes. Meanwhile, prepare an ice-water bath. Heat remaining juice until warm; add to gelatin mixture. Stir until dissolved. Set bowl in ice-water bath; let cool to room temperature, stirring. Slowly pour over pate refrigerate until firm, 3 hours or overnight. Lift up on plastic wrap to unmold. Serve with toast or crackers.