Day 1: Prepare the Dough

1. Mix Dry Ingredients: In a large bowl or the bowl of a stand mixer, combine the bread flour, all-purpose flour, instant yeast, kosher salt, and granulated sugar.
2. Add Wet Ingredients: Add the egg whites and water (or milk) to the dry mixture. Mix until the dough begins to come together.
3. Incorporate Butter: Gradually add the softened butter, one piece at a time, mixing continuously until fully incorporated. Continue kneading until the dough is smooth and elastic, and it pulls away from the sides of the bowl.
4. First Rise: Shape the dough into a ball and place it in a lightly oiled bowl. Cover with plastic wrap or a damp cloth and let it rise in a warm place for about 90 minutes, or until doubled in size.
5. Chill Dough: After the initial rise, gently deflate the dough, shape it into a rectangle, wrap it tightly in plastic wrap, and refrigerate for at least 1 hour or overnight.

Day 2: Lamination and Fillings

Lamination:

1. Prepare Butter Layer: Place the softened butter between two sheets of parchment paper and shape it into a thin, even rectangle measuring approximately 7×8 inches (18×20 cm). Chill briefly if it becomes too soft.
2. Roll Out Dough: On a lightly floured surface, roll the chilled dough into a rectangle twice the size of the butter layer.
3. Encase Butter: Place the butter layer onto one half of the dough rectangle. Fold the other half over the butter, sealing the edges to encase the butter completely.
4. First Fold (Single Turn): Roll the dough into a long rectangle, then fold it into thirds like a letter. Wrap in plastic wrap and refrigerate for 30 minutes.
5. Second and Third Folds: Repeat the rolling and folding process two more times, chilling for 30 minutes between each fold. After the third fold, wrap the dough and refrigerate for at least 1 hour.

Vanilla Bean Pastry Cream:

1. Heat Milk: In a saucepan, bring the milk and vanilla paste (or extract) to a simmer over medium heat.
2. Whisk Eggs and Sugar: In a separate bowl, whisk together the egg yolk, whole egg, granulated sugar, flour, and cornstarch until smooth.
3. Temper Eggs: Gradually pour a third of the hot milk into the egg mixture, whisking constantly to prevent curdling.
4. Cook Cream: Return the tempered egg mixture to the saucepan with the remaining milk. Cook over medium heat, whisking continuously, until the mixture thickens and comes to a boil. Boil for 1-2 minutes, then remove from heat.
5. Chill: Transfer the pastry cream to a bowl, cover with plastic wrap touching the surface to prevent a skin from forming, and refrigerate until fully chilled.

Strawberry Jam:

1. Macerate Strawberries: In a saucepan, combine the sliced strawberries and granulated sugar. Let sit for 15 minutes to release juices.
2. Cook Jam: Place the saucepan over medium heat and bring to a simmer. Cook, stirring occasionally, until the mixture thickens to a jam-like consistency, about 10-15 minutes.
3. Strain and Chill: If desired, strain the jam to remove seeds. Transfer to a jar and refrigerate until needed.

Day 3: Shaping, Frying, and Assembly

1. Roll Out Dough: On a lightly floured surface, roll the laminated dough to a thickness of about 0.5 inches (1.25 cm).
2. Cut Cronuts: Using a 3.5-inch (9 cm) round cutter, cut out circles of dough. Use a smaller cutter to remove the center of each circle, creating a donut shape. Place the cut cronuts on a baking sheet lined with parchment paper.
3. **Proof Cover the cronuts lightly with plastic wrap and let them proof at room temperature for about 90 minutes, or until they have risen slightly.