How to Make It
Melt 2 tablespoons of the butter in a large skillet over medium heat.
Add the chicken livers and cook until browned but still slightly pink inside, 2 to 3 minutes per side. Transfer to a blender; reserve the skillet.
Add the shallot and thyme sprig to the reserved skillet. Cook until the shallot is soft, 3 to 4 minutes.
Remove the skillet from heat and carefully add the brandy. Return to heat and cook until evaporated, about 1 minute.
Discard the thyme sprig. Add the shallot mixture, cream, 1 teaspoon salt, and ¼ teaspoon pepper to the livers in the blender. Blend until smooth.
Press through a fine-mesh sieve into a small serving dish.
Melt the remaining tablespoon of butter and pour over the mousse. Chill until set, 1 to 2 hours.
Serve chilled or at room temperature, topped with the thyme leaves and with the toast on the side.