Set a medium saucepan of water on the stove to simmer. Cut broccoli into large pieces where there are natural separations, making sure the pieces are all the same size. Blanch broccoli by placing into simmering water for 1-2 minutes. Test broccoli for doneness by sticking a paring knife or cake tester into a stalk: it should feel "just below completely tender." Using a slotted spoon, remove broccoli and spread out on a rimmed baking sheet to cool to room temperature.
For the misonnaise: Add the egg, lemon juice, and red miso to the blender and blend, starting on a low speed and gradually increasing to medium; add half the vegetable oil in a slow stream while the blender runs. Once there's a basic emulsion, add water; then add the rest of the vegetable oil in a slow stream while the blender continues to run. Place the misonnaise in a bowl and refrigerate to settle for a few minutes or up to several hours.
For the broccoli: Heat half of the olive oil in a cast-iron skillet over medium heat. When the olive oil is smoking hot, carefully add broccoli to brown; add more oil as it cooks. Place charred broccoli back on the rimmed baking sheet to cool. Zest the lemon over the broccoli. In a large serving bowl, place a few heaping tablespoons of misonnaise on one side. Pile the charred broccoli on the other side and serve.