If you plan on canning the finished jam, prepare a boiling water bath canner and three half pint jars. Wash three rings and lids with warm, soapy water and set them aside.
Chop the tomatoes. Place them in a roomy bowl and add sugar. Stir to combine. Let sit for at least ten minutes, until juicy. If you prefer, you can cover the tomatoes and refrigerate them for up to 24 hours before proceeding with the jam.
When you're ready to cook, pour the sugared tomatoes into a low, wide, nonreactive pan, like a large stainless steel skillet and place it on the stove. The large amount of surface area allows the tomatoes to cook down quickly. If you use a narrower cooking vessel, the cooking time will increase.
Add the lemon juice (if you plan on canning the finished product, please make sure to use bottled lemon juice, as that ensures that the finished product contains enough acid to ensure safety), ginger, red pepper flakes, smoked paprika, salt, and cayenne. Stir to combine.
Turn the burner to high heat and bring the tomatoes to a rapid boil. Reduce the heat to medium high and cook, stirring often, for 15-20 minutes. You want to cook until the tomatoes no longer look watery and have taken on a thick, glossy finish. My favorite way to judge whether a skillet jam like this one is done is by watching how it behaves as I stir. When it is nearly done, you'll be able to clear a spot in the jam with your spoon and it will stay open for a moment or two, before the fruit rushes back in to fill it up. I like to say that it's nearly sculptable when it's done.
When you judge that the jam is done, remove the pan from the stove. Funnel the finished jam into the prepared jars. Wipe the rims, apply lids and rings, and process in a boiling water bath canner for 15 minutes.
When the time is up, turn off the heat and remove the lid. Let the jars rest in the cooling water for 5 minutes. When that time has elapsed, remove the jars and set them on a folded kitchen towel to cool. When the jars have cooled enough that you can comfortably handle them, check the seals. Sealed jars can be stored at room temperature for up to a year. Any unsealed jars should be refrigerated and used promptly.