Make the Caramelized Onions (note: makes about 3 cups): Heat the oil in a wide Stainless Clad Rondeau or Enameled Cast Iron Dutch Oven over medium-high heat. Add the onions, salt, pepper, thyme, and rosemary, stir to mix with the oil, then cover and cook, undisturbed, for 5 minutes. Lower the heat to medium, uncover, and stir. Continue cooking, stirring occasionally, until deep golden brown and soft, 30–40 minutes. Add the vinegar and continue cooking, stirring to scrape up any browned bits on the bottom of the pan. Remove from the heat and let the onions cool completely. Transfer to an airtight container and refrigerate for up to 10 days.

Make the Fancy Cheese Sauce (note: makes about 6 cups): In a large bowl, combine the Gouda and cheddar with the cornstarch and toss to combine. In a medium Stainless Clad Saucier or Saucepan, combine the milk, cream, sodium citrate, salt, nutritional yeast, mustard powder, and paprika and bring to just below a simmer over medium heat. While whisking, begin adding the cheese mixture, one small handful at a time, stirring until it melts into the mixture before adding more. Continue cooking, whisking often, until all of the cheese mixture has been added and the sauce is bubbly, melted, and smooth. Stir in the Parmigiano until fully incorporated, then remove the pan from the heat and let cool. Transfer to an airtight container and refrigerate for up to 1 week.

Make the Cherry Pepper Relish (note: makes about 1 ½ cups): In a blender, combine the cherry peppers, chile paste, vinegar, oil, agave syrup, cayenne, and salt. Blend on low speed until smooth-ish, with small bits of texture. Transfer to an airtight container and refrigerate for up to 2 weeks.

Make the Sandwiches: Take the sliced beef and slice it further into ½-inch (12 cm) strips. Season the sliced beef with salt and pepper.

In a small bowl, stir together the butter and liquid shio koji. Split each bun horizontally, leaving it hinged on one side, and open it up. Spread the shio koji butter evenly over the insides of all the buns.

Heat the oil on a Carbon Steel Griddle or large CeramiClad™ Frying Pan over high heat. Add the beef (work in batches, if needed) and cook until well browned and mostly cooked through, 3–4 minutes. Add 1 ⅓ cups caramelized onions and warm through with the beef. As soon as the beef and onions are cooked, directly transfer them from the griddle to the buttered buns using a metal spatula. Spoon about ½ cup warm cheese sauce per sandwich over the beef and drizzle with a spoonful of the cherry pepper relish. Garnish with chives batons and serve warm.