You mentioned breakroom. Is this a large room or small meeting room? You have no opportunity to sit beside another female co-worker and have lunch on the same table?
I always eat solo and have a special table where I'll bring out a Tablet, some days bring popcorn, and watch something on the Tablet and have a TV-dinner/or cinema experience with the popcorn in the middle of the work-shift. If I see there is a group of women chatting something interesting, I'll pause what I'm looking at, take off the headphones, and start listening to what is going on instead. If I feel a good vibe, I might get off from that table and approach them. If not, I might just stay there. There is a gay guy in the cafeteria that seems to dominate the social scene and talk with the women there (don't worry, I don't like most of the women so I don't feel that bad).
However, I might get a nice hi or greeting from a co-worker on route to the cafeteria or returning from the cafeteria, so it's supplements interactions that might be missing there. Generally, I think I'm the only guy that's doing that with a Tablet...which is like a smartphone on steroids, especially if you add popcorn, bluetooth headphones, and Tablet stand to the mix.
That being said, I do have a special playlist on youtube that consists of vlogs on different topics by women. If I want to feel like I'm eating with a woman, there is a variety of content on youtube that can mimic an experience of having lunch with someone. However, I have not recently felt triggered enough by anything in the cafeteria environment to go that far.
At least what you are doing does not sound common in my workplace....or maybe I'm a bit more extreme in creating a deeper solitary experience than average to compensate for the lack of social connection.
There you go. Try having a chat with a female co-worker in one of those places. The ones who are friendly with you and don't mind being in the same space. If a woman is willing to share a space like that with you, then she must be interested.