Source: Daniel Gritzer, seriouseats.com

Ingredients

2 tablespoons mirin (sweet rice-based wine, it can be found in any Japanese or Asian grocery store.)
2 tablespoons sake
1/4 cup mild red or white miso
6 tablespoons sugar, divided
1 large globe eggplant, cut crosswise into 4 (1/2 inch thick) rounds (see note above)
Vegetable oil, for brushing
1/2 cup rice vinegar
1/2 cup water
1/2 tablespoon kosher salt, plus more for seasoning
2 cups thinly sliced cucumbers
1 1/2 pounds freshly ground beef chuck, preferably about 80 percent lean, formed into 4 (4 1/2 inch wide) patties
Freshly ground black pepper
Mayonaise, preferably Kewpie, for serving
Sriracha, for serving
4 English muffins, split and toasted

Directions

Preheat oven to 350°F. In a medium bowl, whisk together mirin, sake, miso, and 2 tablespoons sugar until well combined. Set aside.

Line a rimmed baking sheet with parchment paper. Brush eggplant slices with vegetable oil and set on prepared baking sheet. Bake eggplant, turning once, until very tender, about 25 minutes.

Meanwhile, combine vinegar, water, remaining 4 tablespoons sugar, and 1/2 tablespoon salt in a small saucepan and bring to a boil over high heat. Remove from heat. Add cucumber slices and stir. Cover with a clean kitchen towel or a double layer of heavy duty paper towels, pressing down until towel is saturated with liquid and in direct contact with the cucumbers. Let rest 10 minutes, then transfer cucumbers to a sealable container. Top up with liquid and discard the rest. Store in the refrigerator for up to 1 month.

Season patties generously with salt and pepper. Cook to desired doneness on a hot grill or in a skillet on the stovetop, 125°F for medium rare or 135°F for medium. Transfer to a plate and allow to rest for 5 minutes.

Turn on broiler. Brush top sides of eggplant slices with miso glaze and broil until browned on top, about 2 minutes. Spread mayonnaise and sriracha on English muffin top halves. Set burger patties on English muffin bottom halves, and top each with an eggplant slice. Top with pickled cucumbers, saving the rest for another use, close burgers and serve immediately.