Position 2 oven racks in the upper and lower thirds of the oven, and preheat to 375 degrees F. Line 2 baking sheets with parchment.
Put the green beans in a large microwave-safe bowl with a splash of water, cover and microwave until bright green and crisp-tender, about 5 minutes. Let cool, then toss with the oil and a large pinch of salt.
Meanwhile, whisk together the flour, baking powder, baking soda and 1 teaspoon salt in a medium bowl. Add the butter, and work it into the flour mixture with your fingers until the butter is broken down into very small pea-size pieces. Make a well in the center, and pour in the milk. Stir the dry ingredients into the wet with a fork.
Once the dough comes together, knead it gently a couple of times on a well-floured work surface. Roll it out to a little thinner than 1/8 inch. Cut it into as many 1 1/2-by-3-inch rectangles as you can; discard any scraps. On each rectangle, lay 4 green bean halves crosswise (the tips should poke out a little) and top with about 1 teaspoon of the cheese; roll the rectangle up, making sure the ends overlap; and seal and arrange, seam-side down, 2 inches apart on a prepared baking sheet. Brush each with egg, and sprinkle with caraway seeds.
Bake until golden brown, 15 to 18 minutes. Let cool on the baking sheets for a few minutes.
Meanwhile, mix together the mustard and sour cream in a small bowl. Serve on the side as a dipping sauce.
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