Cook pasta in a large pot of generously salted water until very al dente. Reserve 1 cup of pasta water before draining.
Season livers with salt and pepper. Melt 2 tablespoons butter in a large saute pan over medium-high heat. Cook livers, flipping once, until plump and golden brown, about 4 minutes; transfer to a cutting board. Reduce heat to medium and melt remaining tablespoon butter. Add onion and 1/4 teaspoon salt; cook until golden, about 8 minutes. Chop livers and return to pan, along with any juices and sage. Increase heat to medium-high. Add wine; simmer until liquid is reduced by half.
Add pasta and 1/2 cup reserved pasta water. Simmer until sauce coats pasta; remove from heat. Toss with Parmesan and parsley. If dry, toss with more pasta water, a bit at a time. Divide among 6 bowls; drizzle with oil and sprinkle with Parmesan.