Heat oil in a large skillet over medium heat. Add onion; cook until soft and lightly browned, about 8 minutes. Add shredded meat to pan; toss to combine. Remove from heat; let cool to room temperature. Meat mixture should be moist, but drain excess liquid.
Place a wonton wrapper on work surface. Top with about 1 teaspoon filling. Using a pastry brush, lightly brush the edges with water. Fold wrapper over, forming a triangle, to enclose filling.
Transfer ravioli to parchment-lined baking sheet in a single layer. Freeze pasta on baking sheet until firm (ravioli will keep, frozen, for up to 2 weeks) so it is easier to handle.
Bring a large pot of water to a boil; add salt. Reduce to simmer; cook pasta until al dente, about 2 minutes. Reserve a 1/2 cup cooking liquid; drain pasta. Set aside.
In a large skillet, heat 2 tablespoons butter over medium-low heat; stir in horseradish. Add ravioli to large skillet; toss gently, adding some reserved cooking water to loosen, if necessary. Taste for horseradish flavor, adding more if desired. Transfer to a serving platter.
Meanwhile, in a small 6- to 8-inch skillet, heat remaining 4 tablespoons butter over medium heat. Increase heat to high; cook until butter just turns brown, being careful it does not burn. Drizzle browned butter over ravioli; garnish with chives, fresh horseradish, and cheese. Serve immediately.