Bring 7 cups of water and salt to a boil in a large pot.
Add cornmeal in a slow stream, stirring continuously to avoid lumps.
Reduce the heat to low and cook until the mixture thickens and the cornmeal is tender, stirring often, about 15 minutes. If needed, add hot water to keep the polenta soft enough to stir. Add up to 1 additional cup.
Turn the heat off and add butter. Stir to combine. Check the thickness of your polenta by scooping some up with a spoon. If the polenta is thick enough to stick to the spoon when you turn the spoon upside down, then it is ready! If the polenta is still a bit watery and thin, continue cooking on low heat and stirring until thick. (If the polenta is too watery and thin, it will be difficult to do the next step.)
Pour onto a flat surface (cookie sheet or large cutting board). Use the back of the wooden spoon to smooth the corn mush to an even thickness.
Refrigerate the polenta for 30 minutes, or cover tightly and refrigerate up to 2 days.
Remove the polenta from the refrigerator. Use cookie cutters to stamp out shapes or cut into squares.
Heat 1 Tbsp. of butter or oil in a frying pan over medium heat and cook the polenta shapes for 3-5 minutes on each side or until golden. You can also serve these cold.
Serve polenta shapes with the jam as a sweet dipping sauce. You can heat the sauce and add ~1Tbsp. water to make it a thinner consistency.