In a medium saucepan, melt 1 tablespoon of the butter. Add the barley and cook over moderately high heat, stirring until fragrant and lightly browned, about 5 minutes. Add 3 cups of water and 1 teaspoon of salt and bring to a boil.
Cover and cook over low heat until the barley is tender and the water is absorbed, about 20 minutes.
Meanwhile, in a large deep skillet, bring 1/2 inch of water to a boil. Add the broccoli florets, cover and cook just until bright green, about 2 minutes.
Drain and pat dry. Wipe out the skillet and melt 1 tablespoon of the butter in the olive oil.
Add the broccoli stems and stir-fry over high heat until crisp-tender, about 2 minutes. Add the shallots, ginger and jalapeño and stir-fry for 1 minute.
Add the broccoli florets, season with salt and stir-fry for 2 minutes.
Off the heat, stir in the barley and scallion and the remaining 1 tablespoon of butter. Transfer to a bowl and serve warm.