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1
In a large saute pan, heat the olive oil and butter over medium-high heat.
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2
When the butter is melted and beginning to fizz, add the onion, salt, and pepper and cook until it's just beginning to soften, 3 to 4 minutes.
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3
Add the diced sweet potato to the pan.
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4
Without stirring, cook the sweet potato until the first side is just browned, about 4 minutes.
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5
Toss and stir well, redistributing, and cook for another 2 minutes without stirring.
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6
Add the apple to the pan and cook, stirring often, until it's just softening and is warmed through.
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7
By adding ingredients to the pan slowly, you'll get sweet and nicely caramelized onion, crispy sweet potato, and just-cooked apple that will still maintain its shape and texture.
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8
Reduce the heat to medium-low and make four 2-inch-wide wells in the saute pan.
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9
To do this, push the vegetables aside with a spoon.
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10
Make sure the pan still has a bit of grease (add some butter to each well, if needed), crack 1 egg into each well, and cover the saute pan.
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11
After 2 minutes, remove the lid and stir the sweet potato and apple without disturbing the eggs.
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12
Replace the lid and cook until the eggs are done to your liking; another 2 minutes (for an easy yolk), or up to 6 minutes (for a hard yolk).
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13
Serve immediately alongside toast.
