Make the dough: Whisk 1/2 cup flour with warm water and yeast in the bowl of a stand mixer. Let sit until foamy, 10 to 15 minutes.

Stir in 2 tablespoons oil, egg white, sugar, salt, and honey; whisk until smooth.

Change the mixer attachment to the dough hook. Gradually add remaining flour to yeast mixture with the mixer running; knead until a soft, sticky dough forms, 3 to 4 minutes.

Drizzle remaining vegetable oil into the mixer 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 and press into a rectangle. Cut into 12 pieces and form each piece into a roll. Place rolls 2 inches apart onto the prepared baking sheet. Cover and let rise until doubled in size, about 30 minutes.

Preheat the oven to 425 degrees F (220 degrees C).

Make the topping: Whisk egg white and 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.