Bring a large pot of salted water to a boil over high heat. Add the pasta and cook until tender but still firm to the bite, stirring occasionally, about 8 to 10 minutes. Drain and reserve about 1 cup of the pasta water.
In a 12-inch skillet, heat the oil over medium heat. Add the garlic and cook until fragrant and golden, about 1 minute. Remove the garlic and discard.
Add the treviso, spinach, chicken stock, balsamic vinegar, lemon juice, 1 tablespoon of salt, and red pepper flakes. Cook until the treviso and spinach until wilted, about 6 to 8 minutes.
Add the pasta and Parmesan. Toss well and thin out the sauce with a little pasta water, if needed. Season with salt, to taste.
Transfer the penne to serving bowls. Top each portion with the crumbled goat cheese and garnish with basil before serving.