Mix the dill or sour-pickle juice, sugar, rye flour, and instant yeast into a soft, thick batter. Cover and let sit for 20 minutes to hydrate the rye.

Add the sour cream or Greek yogurt, caraway seeds, salt, all-purpose flour, and vital wheat gluten if using. Knead until fairly smooth. The dough will naturally be sticky; avoid adding excess flour just to make it behave like white bread dough.

Place the dough in a lightly oiled bowl, cover, and let rise for 1 to 1 1/2 hours, until noticeably puffy.

Gently deflate the dough, knead very briefly, and shape into one long oval loaf. Place on parchment, cover, and proof for about 1 1/2 hours, until noticeably puffy. Near the end of the proof, preheat the oven to 350°F (175°C).

Lightly mist or brush the top with water. With a very sharp knife or razor held fairly upright, make one confident lengthwise slash about 1/2 inch (1.25 cm) deep.

Bake on the center rack. Check the loaf after 35 minutes for color and internal temperature, then continue baking as needed, about 45 to 50 minutes total for one large loaf. Aim for an internal temperature of 205–210°F (96–99°C). If the loaf is getting too dark, loosely tent it with foil after about 25 minutes.

Transfer to a rack and let cool before slicing. For a softer crust, optionally brush the hot loaf with melted butter.