Stir together broth, soy sauce, mustard, miso paste, and red pepper in a small bowl.

Cut mushrooms in half lengthwise. Score the inside of each mushroom half with shallow diagonal cuts (do not cut all the way through).

Heat vegetable oil in a large skillet over high heat, swirling to coat the bottom. Cook mushrooms, cut sides down, until bottoms are golden brown, 3 to 5 minutes. Reduce heat to medium and pour half of broth mixture into the skillet. Cook, covered, until liquid is mostly absorbed and mushrooms are fork-tender, about 5 minutes. Transfer mushrooms to a bowl.

Add remaining oil and shallot to the skillet. Cook, stirring constantly, over medium heat for 30 seconds. Add sake and bring to a boil over high heat. Cook, stirring frequently, until slightly reduced, about 1 minute.

Add coconut milk, remaining broth mixture, and any accumulated juices from mushrooms. Cook over high heat just until the sauce starts to thicken, about 2 minutes.

Return mushrooms to the skillet. Reduce heat to low, simmer until mushrooms are heated through, adding water as needed until the sauce reaches the desired consistency, about 2 minutes.