Step 1

Working in batches, pulse button and shiitake mushrooms in a food processor until finely chopped. Transfer to a medium bowl.

Step 2

Working in batches, pulse tofu in food processor until pea-size. Transfer to a small bowl.

Step 3

Pulse onion in food processor until very finely chopped. Transfer to another small bowl.

Step 4

Heat ¼ cup oil in a large pot over medium. Cook mushrooms, stirring occasionally, until browned and shrunken by about one third, 8–10 minutes. Transfer mushrooms back to their bowl. Wipe out pot.

Step 5

Heat remaining 3 Tbsp. oil in pot over medium. Add onion and garlic and cook, stirring occasionally, until onion is tender and browned around the edges, 8–10 minutes.

Step 6

Add gochujang, five-spice, cumin, and tofu to pot and cook, stirring, until tofu and onion are coated, about 2 minutes. Pour in wine and cook just until alcohol smell has burned off, about 1 minute. Add oyster sauce and 4 cups water and return mushrooms to pot. Partially cover pot and cook mushroom mixture until mixture is thickened and very soft and silky, 50–65 minutes. Remove from heat and season with salt.

Step 7

Whisk vegetable oil and sesame oil in a small bowl to combine, then brush cut sides of each English muffin with oil mixture.

Step 8

Heat a large nonstick skillet over medium-high. Cook English muffins, cut sides down, until golden brown underneath, about 1 minute. Transfer to plates, arranging cut side up.

Step 9

Spoon some mushroom mixture onto bottom halves of English muffins (you may have a bit extra) and top with scallions, cilantro, and sesame seeds. Close up sandwiches.