This could be a "shime" 締  ending dish or, more traditionally, a breakfast. It is a variation of "
Okayu" お粥 or porridge but in a US Southern style. It  is a very comforting and nice starch dish. This one is prepared by my  wife and I only help by chopping up the chives and making the poached  eggs.
Grits is a very  Southern food and my wife is not even from the South but we adapted this  dish for some reason. Instead of the more traditional grits which is  eaten as a breakfast with additions of cheese, butter or maple syrup (a  Northern addition to a Southern dish...don't ask why), we make it  savory. We cook the grits in milk instead of water. Just in case you are  not familiar with grits, it is coarsely ground alkaline-treated hard  kernel corn. Alkaline treated corn prevents the development of a niacin  deficiency (
Pellagra) especially if your main diet consists of  corn. I do not know who first figured this out but certainly American  Indians ("
hominy" from which grits is made) and Mexicans ("
masa  harina" for corn tortilla is also made from alkaline treated  corn) must be among the first. This treatment also created another  flavor different from regular corn meal. Grits and polenta are similar  yet distinctive in their taste. The description below is how my wife  makes grits.
 
Grits: She prefers to use "Quick" grits (Quaker  brand) rather than "Instant" grits. Quick grits, which has been  pre-steamed and then dried, requires about 5 minutes cook time but it is  more "authentic" in flavor and consistency than  the "instant" variety.  True old fashioned stone-ground grits will take way too long to cook. 
 
For  two small serving, in a small but deep pan, add milk (one cup, we use  1%) and just when it starts boiling, add salt (1/4 tsp) and grits (3  tbs) while stirring with a whisk, turn down the flame and cook covered  for 5 minutes until all the milk gets absorbed. Add chopped fresh chives  (3-4 tbs). She adds 1 pat of  butter (1/2 tbs) but no cheese.
Poached  eggs: I poach eggs very simply without any vinegar and without any  special techniques or gadgets but works all the time. I used a 8 inch  frying pan with non-stick surface and with a high side wall so that it  can have enough depth of water to completely submerge the eggs. After  the water comes to a gentle boil, I adjust the heat so no or only few  bubbles form in the water (many bubbles result in egg-drop soup). I then  add 1 tsp of salt (just for seasoning). I crack the egg into a small  ramekin or bowl and gently slide the egg into the water. Repeat the  process. After the white is set and the egg yolk becomes opaque on the  surface (4-5 minutes), using the perforated spoon or silicon spatula  gently pry the bottom of the eggs from the surface of the pan; take care  not to break the yolk. The eggs should come loose without a problem  because of the non-stick surface. Pick up the egg using a perforated  spoon and drain on a paper towel-lined plate.
Place the  grits in a bowl and put the poached egg on top. Sprinkle with salt and  serve. For breakfast my wife serves this with her home-made buttermilk  biscuit (another Southern specialty) but for the "shime" dish serve as  is. Break the egg yolk and mix it with the grits. Even if you are not  from the South, you will like this dish.