Soak the chire in 1/2 cup of milk for about 15-20 minutes till soft.
Add plain flour to the soaked chire, knead to make a soft dough. (If the chire is a bit too mushy, dont hesitate to add a bit more of flour, just enough to form the dough.) Refrigerate.
Add the grated coconut and 3/4th of the jaggery to a pan and cook over a low flame, stirring frequently. Till the jaggery is well incorporated into the grated coconut. 5-7 minutes I would reckon. Remove from heat.
Allow the coconut-jaggery mixture to cool just enough for you to handle it. Now take 1 tbsp of the mixture in your palm, give it an oval shape. Repeat for the rest of the mixture. Keep aside.
Take the chire dough out of the refrigerator. Shape small balls of the dough, keep aside.
Now take a dough ball on your palm, press it gently in the middle using your thumb forming a small oval cup. Place a coconut-jaggery oval ball in the middle of the cup. Very gently bring the edges of the dough cup together to cover the stuffing. Using your palms, smoothen to a nice oval shape. Repeat for the rest of the dough.
Heat oil in a deep-bottomed pan, deep fry the pulis till golden brown. Keep aside on a kitchen absorbent towel.
Pour the milk into a heavy bottomed saucepan, let it simmer over a low flame till it reduces to almost half in volume.
Once the milk has reduced, add the remaining grated jaggery. Give it a hearty stir, continue to cook till the jaggery has melted. (If you like it sweeter, feel free to indulge - add some more grated jaggery !!)
Gently add the chire dumplings one by one into the milk. Cook over a medium flame for about 5-7 mins. Carefully stir the milk a couple of times in between.
Allow to cool to room temperature and serve.