In a large mixing bowl, whisk together the active sourdough starter and water until mostly dissolved.
Add the bread flour and whole wheat flour to the bowl and mix with a dough scraper or hands until no dry flour remains and a shaggy dough forms.
Cover the bowl and let the dough rest (autolyse) at room temperature for 45–60 minutes.
Sprinkle the salt and optional pickled jalapeño brine over the dough and gently pinch and fold the dough until the salt is fully incorporated.
Cover the bowl and let the dough rest 20 minutes at room temperature.
Perform the first set of stretch-and-folds by pulling one side of the dough up and folding it over itself, repeating on all four sides.
Cover and rest the dough for 30 minutes at room temperature.
Perform a second set of stretch-and-folds, then gently fold in half of the jalapeños and cheddar cubes.
Cover and rest the dough for another 30 minutes at room temperature.
Perform a third set of stretch-and-folds, folding in the remaining jalapeños and cheddar cubes as evenly as possible.
Cover and let the dough bulk ferment at room temperature for 3–5 hours, or until it has risen by about 50% and looks airy with bubbles.
Lightly flour a work surface and gently turn the dough out of the bowl without deflating it completely.
Pre-shape the dough into a loose round by folding the edges toward the center, then flip seam-side down and tighten into a ball using a bench scraper.
Cover the pre-shaped dough with a towel and let it rest for 20–30 minutes.
Lightly flour a proofing basket or bowl lined with a towel, using rice flour if available to prevent sticking.
Perform the final shaping by flipping the dough seam-side up, folding the edges toward the center to create tension, then shaping into a tight round or oval.
Place the shaped dough seam-side up in the prepared proofing basket or bowl.
Cover tightly with plastic wrap or a reusable cover and refrigerate for 10–12 hours for a cold retard proof.
About 45 minutes before baking, place the Dutch oven with its lid in the oven and preheat to 475°F (245°C).
Cut a piece of parchment large enough to fit under the dough and invert the chilled dough onto the parchment, now seam-side down.
Score the top of the dough with a sharp knife in a cross or single long slash to allow for controlled expansion.
Carefully remove the hot Dutch oven from the oven, lift the dough on the parchment, and lower it into the Dutch oven.
Cover with the lid and bake for 20 minutes at 475°F (245°C).
Remove the lid, reduce the temperature to 450°F (230°C), and bake for another 15–20 minutes, or until deep golden brown and internal temperature reaches about 205°F (96°C).
Carefully remove the bread from the Dutch oven and transfer it to a wire cooling rack.
Let the bread cool completely for at least 1 hour before slicing to allow the crumb to set and cheese to firm up.