Heat olive oil in a large skillet over medium-high heat. Add the sausage and cook until the sausage is no longer pink, 3 – 4 minutes. Add the onion and cook until tender, about 5 minutes. Add the garlic and cook for 30 seconds. Drain off all but 1 tablespoon of the fat.
Add the kale, 1 teaspoon salt, ½ teaspoon black pepper, and the red pepper flakes. Reduce heat to medium-low, cover the pan, and cook for 20 minutes or until the kale is tender. Increase heat to high, add the balsamic vinegar and cook, stirring, until the vinegar is evaporated. Remove from the heat.
While the sauce is cooking, bring a large pot of heavily salted water to a boil. Cook the linguine according to the package directions, Reserve ½ cup of the pasta cooking water, drain the pasta, rinse with hot water, and add to the pan with the sausage and kale along with ¼ cup of the reserved pasta water. Stir to combine, adding a little more pasta water if the dish seems dry. Adjust seasoning with more salt and pepper is desired.