In a deep, heavy saucepan, melt butter over medium heat and add the onions, celery and basil.
Saute until the celery and onions are just beginning to get tender. Add the water/stock, the potatoes, ham and the pepper. Cook until potatoes are tender.
Mix the cornstarch (the amount is not a typo) with the 3/4 cup water.
Slowly add enough of the cornstarch/water mixture (you may not have to add it all, do it in small steps) to produce a VERY THICK mixture. The spoon should practically stand up in this. It should be the consistency of a very thick pudding. Stir constantly until the mixture bubbles.
GRADUALLY add cream, stirring constantly, to thin the soup out to the desired consistency. You may have to use MORE cream, depending on the desired thickness. I like mine thick, like stew, so the 1 cup of cream works well. Your mileage may vary.
*Add salt if you feel it needs some.
Heat through (do not boil) and serve with a salad or crusty roll. This one's a meal folks, not a starter!!