In a nonreactive bowl, toss the livers, shallot, garlic, chopped sage, thyme and 1 tablespoon of the vinegar; refrigerate for 2 hours.
In a medium skillet, heat 2 tablespoons of the butter. Add the livers and marinade and cook over moderately high heat, turning, until lightly browned, about 8 minutes.
Scrape the mixture into a food processor and pulse until chopped. Add 1 tablespoon of the butter and puree until smooth. Season with salt and pepper. Scrape the puree into a resealable plastic bag and refrigerate until cooled slightly and firm, about 20 minutes. Snip off one corner of the bag.
Arrange 4 wontons on a work surface and brush the edges with water. Pipe a rounded teaspoon of the liver onto each wrapper and fold the wrapper over to create a triangle. Press out any trapped air and press the edges to seal. Transfer the ravioli to a wax paper–lined baking sheet, dusted with flour. Repeat with the remaining wrappers and filling.
Bring a large pot of salted water to a boil. Add the ravioli and cook until al dente, 3 to 4 minutes. Drain and gently shake dry.
Heat the remaining 4 tablespoons of butter in a large skillet. Add the sage leaves and cook over high heat until crisp, 4 minutes. Transfer the leaves to a plate. Add the remaining 1/4 cup of balsamic vinegar to the skillet and simmer until syrupy, 3 minutes. Add the chicken stock and bring to a boil. Add the cooked ravioli and gently stir to coat with the sauce. Transfer the ravioli and sauce to shallow bowls, crumble the fried sage on top; serve.