I am lacto-ovo vegetarian, but I swear by both Lula Cafe (in Logan Square) and Green Zebra ( Ukrainian Village) plus you can take your meat-loving friends and everyone will be happy, at least that has been my experience. You should also check out the Brown Sack ( western-most edge, or just west, of Logan Square).
Victory's Banner is also popular for breakfast/lunch, but it is in Roscoe Village. M Henry has some nicer offerings as well (only breakfast and lunch) but it is on the northern-most edge of Andersonville.
I have not eaten there but Sultan's Market ( Lakeview and I think the original is in Wicker Park) or any other Middle-Eastern restaurant ought to also be of assistance. I don't know of any in the Logan Square area.
One of my favorites is in Lakeview, Yummy, Yummy formerly Mona's vegetarian restaurant, Szechuan Garden, in Chinatown and Lakeview. There is soy gluten for mock meat, seitan as well but I heard you say no wheat. Fried rice made without egg as well as brown rice on offer. I am waiting on a food delivery now. Plus there are delicious bubble teas, no milk, my favorite is mango lychee with the tapioca.
Good Luck!