How to Make It
Preheat oven to 450°F.
Combine oil and salt in a small bowl. Brush oil mixture evenly over both sides of bok choy halves, reserving some oil mixture in bowl. Place bok choy, cut sides up, on a foil-lined baking sheet. Add sliced mushrooms to bowl; toss to completely absorb any oil and salt remaining in bowl. Scatter mushrooms on baking sheet. Bake at 450°F for 8 minutes.
Meanwhile, combine butter, miso, and sugar, stirring well.
Using kitchen shears, cut each lobster tail down the center of the top of the shell, cutting to (but not through) the fan-shaped piece of shell at the end. Turn each lobster tail over, and carefully break the “ribs” on the translucent underside of the shell. Turn lobster tails over; carefully loosen meat from the inside of the shell, being careful of the sharp edges (you might find it helpful to use a spoon to slip between meat and shell). Pull meat up and out of shell, keeping it attached at the bottom at the fan-shaped flipper. Remove and discard the vein, if there is one. Arrange meat on top of shell.
Brush lobster evenly with butter mixture. Arrange on baking sheet with bok choy and mushrooms; turn bok choy over, cut side down. Bake at 450°F until lobster meat is opaque and cooked through, about 10 minutes. The lobster is done at this point, but you can broil it for a minute or two, if desired, to brown the top. Divide rice evenly between 2 plates; divide bok choy and mushrooms evenly between servings. Arrange 1 lobster tail on each plate.