Preheat oven to broil. Lightly oil a baking sheet or coat with nonstick spray.
In a large pot of boiling salted water, cook potatoes until parboiled for 12-15 minutes; drain well.
Place potatoes and broccoli in a single layer onto the prepared baking sheet. Add olive oil, garlic and thyme; season with salt and pepper, to taste. Gently toss to combine.
Season steaks with salt and pepper, to taste, and add to the baking sheet in a single layer.
Place into oven and broil until the steak is browned and charred at the edges, about 4-5 minutes per side for medium-rare, or until desired doneness.