1. Hard-boil the eggs: place eggs in a saucepan, cover with cold water by 1 inch (2.5 cm), and bring to a boil over medium-high heat. Once boiling, remove from heat, cover, and let stand 12 minutes. Transfer to an ice bath for 5 minutes, then peel.
2. Make a thin white sauce (béchamel): melt the butter in a small heavy saucepan over medium heat until foamy, 1–2 minutes. Whisk in the flour and cook, stirring constantly, for 1 minute until the paste smells faintly nutty but has not browned — this step (cooking butter and flour together briefly before adding liquid) is called making a roux. Slowly pour in the whole milk while whisking to prevent lumps. Add the salt and pepper. Cook, stirring, until the sauce thickens enough to coat a spoon and reaches approximately 180°F (82°C), 4–6 minutes total.
3. Separate the hard-boiled egg yolks from the whites. Finely chop the whites and stir them into the finished white sauce.
4. Cut 4 of the 5 toast slices in half lengthwise to make 8 fingers. Arrange them in a single layer on a serving platter.
5. Pour the egg-white sauce evenly over the toast fingers.
6. Press the egg yolks through a potato ricer or fine-mesh strainer directly over the sauced toast, creating a golden, crumbled coating that resembles goldenrod blossoms.
7. Cut the remaining slice of toast into triangular points. Garnish the platter with the toast points and fresh parsley sprigs. Serve immediately while the sauce is hot and the toast retains some crispness.
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