Using a spiral vegetable slicer or julienne vegetable peeler, cut sweet potatoes lengthwise into long, thin strands. You should have about 12 cups of “noodles.”
Cook the sweet potatoes in the boiling water, gently stirring once or twice, until just starting to soften but not completely tender, 11/2 to 3 minutes. Reserve 1/4 cup of the cooking water, then drain.
Return the noodles to the pot, off the heat.
Combine eggs, Parmesan, salt, pepper and the reserved water in a bowl; pour over the noodles and gently toss with tongs until evenly coated.
Heat oil in a large skillet over medium heat.
Add bacon and mushrooms. Cook, stirring often, until the liquid has evaporated and the mushrooms are starting to brown, 6 to 8 minutes.
Add garlic and cook, stirring, until fragrant, about 1 minute.
Add spinach and cook, stirring, until wilted, 1 to 2 minutes.
Add the vegetables to the noodles and toss to combine. Top with a generous grinding of pepper.