To meet your need for foods grannie cooked, there are two I'd particularly recommend.
The first is The Settlement House Cookbook. The Settlement House existed, way back when, to help waves of immigrant women assimilate. The cookbook was part of that process, and taught thousands of them how to cook using American ingredients and methods. As such, it is, perhaps, the quintessential American cookbook.
Second, and as an interesting complement to, and counterpoint, is Molly O'Neil's New York Cookbook. This is a rather unique approach, in which the immigrant recipes are themselves presented---sometimes by the emmigrant him/herself, sometimes by their children or grandchildren, sometimes by restauranteurs who are trying to maintain the old-world foodways. As such, it represents the culinary melting pot that is New York, in particular, and America in general.
After that we get into the country-specific titles you asked for. The list of them goes on and on, and I'll leave it to others to state their opinions. I would suggest, however, that you explore Hippocrene Books' list, as they specialize in ethnic cookbooks written by people who are both cooks and natives of the countries involved, with an emphasis on home cooking.
They have taken the oath of the brother in blood, in leavened bread and salt. Rudyard Kipling