In a bowl place the egg, flour, salt and water. Knead about 8-10 minutes till you get a firm and smooth dough. Cover it with a moist towel and rest it 15-20 minutes.
While the dough is resting, mix all the filling ingredients in a plate and set a side to use at making Mantı.
Divide the dough into 2 or 3 pieces. Take one dough and place on the floured counter. Don't forget to cover rest of the dough. Then flatten the dough with a flat surface rolling pin as thin as you can.
Cut the flatten dough with knife into 3/4 inch (about 2 cm) square pieces. Then use 1/4 of a teaspoon to fill each squares. Then stick the opposite corners of the each square by pressing with your finger tips. Do the same steps for remaining dough pieces.
Boil the water for cooking in a big pot and add salt in it. Then add the all Mantı into the boiling water and stir in every 5 minutes to prevent the Mantıs to stick each other. Do not close the lid at cooking this dish. Cook at medium-high heat about 10-15 minutes.
While it is cooking, you can able to prepare the both yogurt and oil sauce.
In a bowl pour the yogurt, salt and garlic. Mix them well.
In a small pan; melt the butter first and then add the paprika powder and tomato paste. Cook them about 3 minutes at medium heat.
Drain the cooked manti and transfer it into your serving plates. Then put yogurt sauce over it. And finally pour about two tbsp of oil sauce over.
If it is requested, sprinkle some dried mint and sumac over Kayseri Mantı.