Prepare the Brioche Dough:
In the bowl of a stand mixer, mix flour, sugar, and yeast. Add eggs and warm milk. Mix until a shaggy dough forms. Add salt, then gradually incorporate softened butter, mixing until dough is smooth and elastic (8–10 minutes). Cover and let rise for 1–2 hours or overnight in the refrigerator.
Make the Jam:
In a saucepan, combine rhubarb, strawberries, sugar, and lemon juice. Simmer over medium heat, stirring occasionally, until thickened (15–20 minutes). Stir in vanilla. Cool completely before using.
Shape the Donuts:
Roll out dough to ½-inch thickness. Cut into rounds and place on a parchment-lined baking sheet. Cover and let rise until puffy, about 45 minutes.
Fry the Donuts:
Heat oil in a deep pan to 350°F (175°C). Fry donuts in batches for 1–2 minutes per side, until golden. Remove and drain on a wire rack or paper towels.
Fill the Donuts:
Once cooled, use a piping bag fitted with a round tip to fill each donut with jam through a small hole on the side.
Glaze the Donuts:
Mix powdered sugar, milk, and food coloring until smooth. Dip the tops of each donut into the glaze and let set on a rack.