Wow he's actually very smooth. As a bit of of explanation, first off you have to be aware of the unfair expectations that were (and possibly in some areas still are) held of women regarding eating. Proper ladies were expected in company to eat only small bites, neatly, and delicately with a modest appetite (so no stuffing/gorging). Therefore, she feels compelled to avoid eating a sandwich like that in front of him and feigns fullness as an excuse.
Now him eating her sandwich may seem initially inconsiderate, as she took it, but actually is a consideration. By him eating her sandwich it gives him the excuse to order one for her, and one that is specifically cut to small pieces. This is thoughtful in 2 ways. First it gives her a sandwich in a form in which she comfortably eat it in company (small pieces that can be eaten neatly). Second, by doing it this way he can address the issue of her not being able to eat in a way that doesn't draw attention nor would force her to feel indebted by him simply ordering her another sandwich (why he ate hers, so he's making it up to her rather than doing her a favor).
By modern standards this is very roundabout way of doing things but considering the manners for this general timeline, it's a very smooth way of showing consideration and ingratiating himself to her .