Allow eye fillet steaks to come to room temperature, about 25 mins.
Pre-heat oven to 180°C. Line a rimmed oven tray with baking paper. Slice larger carrots lengthwise in half.
Transfer carrots to lined tray and drizzle with 1 tbsp olive oil. Season with salt and pepper, toss to coat in oil, then arrange in a single layer. Roast carrots for 20–25 mins until golden and tender.
While the carrots are roasting, preheat a heavy-based pan over high heat. Pat steaks dry with paper towel, drizzle with a bit of oil and season with a good pinch of salt and pepper.
Place steaks on the pre-heated pan and reduce heat to medium. Cook 3–4 mins on one side, then flip and cook another 3–4 mins.
Transfer steaks to a clean plate, cover loosely with foil and rest for 5 mins before serving.
Wipe out pan, drizzle with remaining 1 tbsp olive oil, and place over medium heat. Add garlic and cook for 1 min. Add spinach, and a pinch of salt and pepper, and cook a further minute, until just wilted. Remove from heat, stir through lemon zest and juice to taste.
Serve steaks with roasted carrots, spinach and pesto.