Brioche dough: In the bowl of a stand mixer, combine flour, yeast, sugar, and salt.
Mix and form a well in the center.
Add eggs and knead, adding milk as needed, until the dough becomes soft and smooth. This operation will take about 8 minutes.

When the dough is soft and smooth, incorporate the butter, a few pieces at a time. Continue kneading until the butter is well combined.
Place the dough in a large greased bowl, cover with plastic wrap, and let rise at room temperature for 1 hour 30 minutes.

Filling: Place the salmon in a large pot and pour in the fish broth.
Bring to a boil and poach for 15 minutes.
Remove the salmon from the broth and reserve it on a plate.
Remove the skin and reserve.

Bring the fish broth to a boil again (using the same broth used to cook the salmon).
Cook the rice in the broth according to the package instructions (30 minutes for the variety of rice I used).
Meanwhile, sauté the baby spinach in a pan with butter.
Season with salt and pepper and reserve in a bowl.

Add the cooked rice to the spinach mixture, then the two egg yolks, heavy cream, and lemon juice, and mix well.
Taste and adjust the seasoning.

Assembly: Prepare a baking sheet lined with parchment paper or a silicone mat.
On a floured surface, roll out the brioche dough into a rectangle of 14 x 20 inches (35 x 45 cm).
Place the dough on the prepared baking sheet (it will overhang the edges).

Spread half of the rice and spinach mixture along the length, leaving a 2-inch (5 cm) gap at each end, and spread it over a width of 5 inches (12 cm).
Place the salmon fillet on top.
Cut the hard-boiled eggs in half and place them on the salmon.
Spread the remaining rice and spinach mixture on top.

Using scissors, cut the dough on each side of the filling into 3/4 inch (2 cm) wide strips, being careful not to cut too close to the filling.
Fold each strip over the filling, then cross them over the filling (like a Danish braid).

Preheat the oven to 400 F (200 C).
Brush the koulibiac with the beaten egg mixed with milk.
Bake for 45 minutes.
The koulibiac can be prepared the day before; to serve, cover with aluminum foil and reheat at 275 F (135 C) for 1 hour.
Serve, sliced and topped with beurre blanc.

Beurre blanc: In a saucepan, combine the chopped shallots and cover with dry white wine.
Bring to a boil and reduce until almost no liquid remains.
This operation can be done in advance.
When ready to serve, heat the saucepan over medium heat (5 on the dial).
Whisking constantly, add the butter, a little at a time, waiting for the previous pieces to melt before adding more.
The mixture will be creamy and smooth.
Beurre blanc should not be reheated, or it will separate.
Serve immediately with the koulibiac.