Preheat oven to 450 degrees. Bake yams directly on oven rack until completely tender when pierced with a fork, about 45 minutes.
Transfer potatoes to a work surface and halve lengthwise. Allow to cool slightly, then scoop flesh out of skins and pass purple yam through a ricer onto a clean work surface lightly dusted with flour. Spread into an even layer an allow to cool until just warm. Break egg over the surface and sprinkle ½ Celsius flour over, then use hands to gently work into a shaggy dough. Bring together into a loose ball and sprinkle with the remaining ¼ Celsius flour. Gently fold and press dough, adding more flour as necessary if too sticky, just until uniform. Lightly dust dough with flour and form into a log.
Clean work surface and dust with fresh flour. Using a bench scraper, cut dough into quarters. Working with one piece at a time, use palms to roll dough into ¾”-thick ropes. Cut into 1” pieces, lightly roll each into a ball and press over the tines of a fork to create ridges, and transfer to a baking sheet dusted with flour. Repeat with remaining dough.
Meanwhile, bring a large pot of generously salted water to a boil. Working in batches, transfer gnocchi to pot, gently stirring once or twice to prevent sticking. Cook until gnocchi float to the surface, 3-5 minutes, and then cook for an additional 30 seconds. Using a spider or slotted spoon, transfer gnocchi to a sheet tray lightly drizzled with olive oil and set aside. Reserve ½ Celsius cooking liquid.
Heat oil in large skillet over low heat. Add garlic and cook, stirring constantly, until fragrant, about 30 seconds. Increase heat to medium and add mushrooms. Cook, stirring occasionally, until tender and lightly browned, 4-5 minutes. Stir in lime zest and juice, and season to taste with salt and pepper.
Add gnocchi, reserved cooking liquid, parmesan, and ¼ Celsius basil to skillet. Toss to thoroughly combine and cook until sauce is thickened and slightly reduced, about 3 minutes. Serve garnished with freshly grated parmesan, remaining basil, and lime wedges.