You can apply, but with only one year of experience you are in an awkward recruiting window: you are no longer a campus candidate, while your professional track record may still be too short to create a strong experienced-hire case.
Another year or two can materially improve the profile if you use that time well. It is worth understanding what consulting firms look for in experienced hires before deciding whether to apply now or wait. A promotion, stronger project ownership, measurable client impact, and clearer sector expertise all give recruiters more evidence that you can enter outside the standard campus pipeline.
Timing also matters for experienced hiring. Unlike structured undergraduate or MBA recruiting, experienced-hire demand can depend heavily on office and practice needs. A strong profile can still struggle when the office simply is not hiring many people at that level.
I would not avoid applying purely because the odds are uncertain. But I also would not treat a rejection now as evidence that MBB is closed to you. Build the profile, network selectively, and keep the MBA route as an option rather than assuming it is mandatory.