Broccoli stalks have more fiber, while the florets are richer in phytochemicals— so do eat the whole thing!
Preheat oven to 400°F. Pulse bread in a food processor until coarse crumbs form, 6 to 8 times.
Cut broccoli lengthwise into 4 wedges, making sure florets stay attached to the stem. Warm 2 tablespoons of the butter in a small skillet over very low heat until melted, 1 to 2 minutes. Remove from heat; transfer to a small bowl, add Worcestershire sauce, and stir to combine. Brush broccoli evenly with Worcestershire mixture.
Heat oil in a large cast-iron skillet over medium-high heat. Place broccoli quarters in skillet, cut side down. Cook, brushing occasionally with Worcestershire mixture, until caramelized, 4 to 5 minutes per side. Transfer skillet to oven; roast until broccoli is tender, 8 to 10 minutes.
Melt remaining 2 tablespoons butter in same small skillet over medium heat. Add bread crumbs and thyme. Cook, stirring often, until bread crumbs are toasted, 4 to 5 minutes. Remove from heat. Stir in cheese and pepper.
Place 1 broccoli wedge on each of 4 plates; top evenly with bread crumb mixture.