In a bowl, soak the dried mushrooms in 1 1/2 cups of very hot water for 10 minutes. Strain, reserving the soaking liquid, and finely chop them.
In a large high-sided skillet, combine olive oil, onion, garlic, thyme, and sage, then turn the heat to high.
Cook, stirring occasionally, until the onion is translucent, about 5 minutes.
Add the chopped mushrooms, spinach, and a couple pinches of salt and pepper and stir well to coat the vegetables with the oil. Reduce the heat to medium and cook for 5 minutes, stirring occasionally.
Pour in the reserved mushroom liquid and stir. When it comes to a boil, turn off the heat.
Bring a large pot of salted water to a boil. Add the pasta and cook, stirring occasionally, until it's tender, but not quite al dente.
Meanwhile, bring the skillet of vegetables back up to medium heat.
Using a spider skimmer or slotted spoon, transfer the pasta from the boiling water to the sauce in the skillet. Toss well and cook until the pasta absorbs all the liquid. If the mixture looks dry, add 2 tablespoons of the pasta cooking water. The pasta should glisten, but water shouldn't be accumulating on the bottom of the pan.
Add the butter and 2 tablespoons of the Parmigiano-Reggiano and toss. If needed, add another tablespoon of pasta water. Remove the pan from the heat.
To finish, sprinkle with the fontina and the remaining 1 tablespoon Parmigiano-Reggiano.