Preheat your oven to 400°F and line a large baking sheet with parchment paper.
In a small bowl, whisk together the minced garlic, dried rosemary, dried thyme, sea salt, black pepper, and half of the olive oil to form a thick herb paste.
Pat the London broil dry with paper towels and rub the herb paste generously over all sides of the meat.
Place the broccoli florets on the prepared baking sheet, drizzle with the remaining olive oil and a pinch of salt, then toss to coat evenly.
Heat a heavy cast-iron skillet over medium-high heat until very hot, then sear the beef for 3 minutes per side to develop a deep golden crust.
Transfer the seared beef to the baking sheet, nestling it among the broccoli florets.
Roast in the oven for 10 to 12 minutes, or until the internal temperature reaches 130°F for medium-rare.
Remove from the oven and let the beef rest on a cutting board for 10 minutes before slicing thinly against the grain and serving with the lemon-spritzed broccoli.