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Step 1
Preheat the oven to 400 degrees.
Step 2
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.
Step 3
Let the potatoes cool enough to handle, then peel and grate, using the largest grating size.
Step 4
In a large bowl, toss the grated potatoes, olive oil, and salt.
Step 5
Grease the muffin tin with olive oil.
Step 6
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.
Step 7
Use your fingers to lightly press the center of each cup so the potatoes spill over the top a bit.
Step 8
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.
Step 9
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.
Step 10
Return to the oven and bake for 10-15 minutes, depending on how cooked you like your eggs. Let cool slightly, then serve immediately.