Meanwhile, cut the butter into a large skillet over medium heat to melt.
Cook, stirring occasionally, until the butter is golden, about 3 minutes. Add the hazelnuts and cook, stirring occasionally, until the nuts are golden, about 3 minutes.
Add the shallot and sauté until softened, about 1 minute more.
Remove from the heat and carefully stir in 3 tablespoons of the vinegar, ¼ teaspoon of salt and several grinds of black pepper.
When the tortellini are ready, reserve ¼ cup of the pasta water with a measuring cup, then drain the tortellini. Add the reserved pasta water and tortellini to the pan and bring the mixture to a simmer. Add the spinach, handful by handful, tossing and gently stirring until the spinach has wilted and the sauce has thickened enough to coat the pasta evenly, about 2 minutes.