Preheat the oven to 400 degrees.
Bake the russet and sweet potatoes until they’re cooked through but not quite tender, about 20 minutes. The goal is just to pre-bake them a bit, so they should still be pretty firm.
Let the potatoes cool enough to handle, then peel and grate, using the largest grating size.
In a large bowl, toss the grated potatoes, olive oil, and salt.
Grease the muffin tin with olive oil.
Add the grated potato mixture to a 12-cup muffin pan, 1/3 cup at a time. The mixture will shrink significantly when baked, so it’s okay if there’s a lot of mixture in each cup. If you end up with extra potato mixture, feel free to use it to make additional egg cups.
Use your fingers to lightly press the center of each cup so the potatoes spill over the top a bit.
Increase the oven temperature to 425 degrees, then bake until the potatoes are golden brown, about 20-25 minutes. Keep on eye on them so the ends of the potato shreds don’t burn.
Remove from the oven, let the nests cool a bit, then crack one egg into each cup and sprinkle with additional kosher salt to taste.
Return to the oven and bake for 10-15 minutes, depending on how cooked you like your eggs. Let cool slightly, then serve immediately.