Whisk 1/2 cup flour, 1 cup warm water, and yeast together in the bowl of a stand mixer. Let sit until foamy, 10 to 15 minutes.
Stir 2 tablespoons oil, 1 egg white, sugar, salt, and honey into yeast mixture; whisk until smooth. Gradually add flour to yeast mixture while mixer is running with the dough hook attached; knead until a soft, sticky dough forms, 3 to 4 minutes.
Drizzle remaining vegetable oil into the bowl and turn dough to coat. Cover the bowl and let dough rise until doubled in size, 1 to 2 hours.
Line a baking sheet with a silicone baking mat.
Turn dough out onto a lightly floured work surface, flatten dough, and press into rectangle. Cut dough into 12 pieces and form pieces into rolls. Place rolls 2-inches apart on prepared baking sheet. Cover and let rise until doubled in size, about 30 minutes.
Preheat oven to 425 degrees F (220 degrees C).
Whisk 1 egg white and 2 teaspoons water together in a small bowl. Use kitchen shears to cut an X at the top of each roll. Brush rolls with egg white mixture and sprinkle with caraway seeds and salt.
Bake in the preheated oven until golden brown, 18 to 20 minutes.
I like to add about 80% of the flour called for and then continue adding small amounts as the dough kneads until I have the perfect texture. You want a soft, fairly sticky dough that pulls cleanly off the sides of the bowl.
Get the recipe for Chef John’s Beef on Weck sandwiches.
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