Preheat the oven to 350° F. Butter an 8-inch square pan and sprinkle sugar to coat the bottom and sides.
Melt 5 tablespoons of the butter over medium heat until it begins to brown and smell nutty. As soon as you see it browning, remove it from the heat, and set it aside to let cool.
Cream together the remaining 5 tablespoons of room temperature butter with the sugar until light and fluffy.
Add the eggs, one at a time, and the vanilla. Mix well, scraping down the sides of the bowl as you go.
Add the condensed milk and beat until the batter is smooth. Add the 5 tablespoons of melted, cooled butter and fold it in (the batter might look too liquidy—it will come together!).
Add the salt and flour and mix until the batter is just combined.
Pour the batter into the prepared pan. If your jam is very thick, now is a good time to warm it slightly—just enough that it loosens a bit. Spoon dollops of jam over the surface of the pan and use a knife to swirl the jam across the surface of the cake.
Bake for 30 to 35 minutes, or until the cake starts to pull away from the sides of the pan and a tester inserted into the middle comes out clean.
Let the cake cool for at least 20 minutes before cutting and serving, or turning out into a wire rack to cool completely.