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PREPARING THE DOUGH
Line a baking sheet with parchment paper; set aside. Butter a large bowl; set aside. Put yeast, warm water, sugar, egg, and 3 tablespoons butter in the bowl of an electric mixer fitted with the dough hook. Mix on medium speed 1 minute.
Reduce mixer speed to low. Mix in salt, flours, and caraway seeds. Raise speed to medium-high, and beat until dough comes together, about 10 minutes.
Turn out dough onto a lightly floured work surface. Knead until smooth, 15 to 20 minutes, then transfer to buttered bowl.
Cover dough with a clean kitchen towel, and let rise in a warm place until doubled in bulk, about 1 hour.
Using a butter knife, cut dough into 12 equal pieces. Roll into balls, and space 1 1/2 inches apart on prepared baking sheet. Cover dough with buttered plastic wrap. Let rise in a warm place until doubled in bulk, about 1 hour.
BAKING THE ROLLS
Preheat oven to 350 degrees.
Using a sharp knife, cut an X in the top of each roll. Bake rolls until golden brown, about 35 minutes.
Transfer rolls to a wire rack set over a baking sheet, and brush with remaining 3 tablespoons butter.