How to Make It
Stir together Kewpie mayonnaise, ketchup, and yuzu juice in a bowl. Refrigerate until ready to serve. Process canned tomatoes in a food processor until smooth, about 1 minute. Set processed tomatoes aside.
Heat oil in a small saucepan over medium-high. Add 1 pound ground beef, onion, and 2 teaspoons salt; cook, stirring occasionally, until meat is no longer pink and onion is lightly browned, about 7 minutes. Add processed tomatoes, tonkatsu, gochugaru, sugar, coriander, and cumin; stir in sake. Bring to a simmer. Stir together 1/4 cup cold water and flour in small bowl; add to ground beef mixture, and stir to combine. Stir in smoked paprika and 1/2 teaspoon black pepper. Reduce heat to medium-low, and return to a simmer. Cook, stirring often, 10 minutes. Remove from heat.
Preheat grill or griddle to medium (350°F to 450°F). Gently stir together remaining 3 pounds ground beef, remaining 2 teaspoons salt, and remaining 1 teaspoon black pepper in a large bowl until just combined. Shape mixture into 8 (6-ounce) patties, pressing to 1/2-inch thickness. Place on oiled grates, and grill, covered, to desired degree of doneness, about 5 minutes per side for medium.
Place patties on buns. Top each with 1/3 cup chili, 1 tomato slice, 1/4 cup lettuce, and 1 tablespoon mayonnaise mixture.