Caramelize the Onions (Do this ahead of time): In a skillet over medium-low heat, melt the butter. Add the sliced onions and cook slowly for 30-45 minutes, stirring occasionally, until they are deeply browned and jammy. Remove from heat, drain any excess fat, and let them cool completely to room temperature.
In a large mixing bowl, whisk the warm water and active starter together. Add the bread flour and whole wheat flour. Mix until a shaggy dough forms and no dry flour remains. Cover and let rest for 45 minutes to autolyse.
Sprinkle the fine sea salt over the dough. Dimple it in with wet fingers and perform a series of stretch and folds for about 3-5 minutes until the dough builds initial tension. Cover and rest for 30 minutes.
The Lamination Stage: Turn the dough out onto a lightly misted countertop. Gently stretch the dough out into a large, thin rectangle. Scatter the cooled caramelized onions, cubed Gruyère, and fresh thyme evenly across the surface. Fold the dough into thirds (like a letter), then roll it up. Place it back in the bowl, cover, and rest for 30 minutes.
Perform 3 more sets of coil folds spaced 30 minutes apart to build strength and distribute the inclusions.
After the last fold, let the dough rest untouched until it has increased in volume by about 40-50%, shows bubbles on the surface, and feels airy.
Turn the dough onto an un-floured surface. Shape it into a tight, round boule. Crucial step: inspect the outside of the dough and poke any exposed cheese cubes deep into the center so they don't burn. Place seam-side up into a rice-floured banneton.
Cover the banneton securely and transfer to the refrigerator for 12 to 16 hours for the cold retard.
Preheat your oven to 500°F (260°C) with a Dutch oven inside for at least 45 minutes. Turn the dough out onto parchment paper, score it, and transfer to the hot Dutch oven. Drop an ice cube under the parchment for extra steam, cover, and bake for 20 minutes.
Remove the lid, reduce the oven temperature to 450°F (230°C), and bake for an additional 20-25 minutes until the crust is deeply caramelized. Cool completely on a wire rack before slicing.