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1
Fill a large saucepan a quarter of the way full with water and bring it to a simmer over high heat.
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2
Remove from the heat and set aside.
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3
Place the chevre, milk, and mustard in a blender and season with salt and pepper.
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4
Blend on high speed until smooth and fully incorporated, about 1 minute.
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5
To keep the sauce warm, place the blender pitcher directly in the reserved hot water while you cook the vegetables and poach the eggs.
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6
Heat the butter in a large frying pan over medium heat until foaming.
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7
Add the leeks, season with salt and pepper, and stir to combine.
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8
Cook, stirring occasionally, until the leeks have softened and are just beginning to brown, about 5 minutes.
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9
Add the asparagus, season with salt and pepper, and stir to combine.
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10
Cook, stirring occasionally, until the asparagus is crisp-tender, about 4 minutes.
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11
Remove the pan from the heat.
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12
Bring the measured water to a simmer in a medium pot over high heat (the water should be at least 2 inches deep).
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13
Reduce the heat to low to keep the water at a bare simmer.
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14
Stir in the vinegar.
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15
Break 1 egg into a small cup or ramekin.
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16
Gently slide the egg into the simmering water.
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17
Repeat with the remaining 3 eggs, making sure to evenly space them in the water.
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18
Cook until the whites are just set, about 3 minutes.
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19
Lift the eggs out of the water with a slotted spoon.
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20
Place in a bowl of warm water to keep warm.
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21
Toast the bread and coat the top of each slice with a thin layer of butter, season with salt and pepper, and place on serving plates.
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22
Divide the asparagus-leek mixture over the toast.
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23
Using a slotted spoon, remove the eggs from the warm water.
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24
Blot any excess water from the underside of the spoon with a paper towel or kitchen towel and slide the eggs onto the asparagus-leek mixture.
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25
Top the eggs with the desired amount of chevre sauce (you may have some left over to serve on the side).
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26
If desired, sprinkle parsley over the top, then serve immediately.
