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Melt butter in a large heavy-bottomed stainless steel or enameled cast iron saucepan over high heat. Add onions and cook, stirring frequently, until bottom of saucepan is glazed in a pale brown fond, about 5 minutes. Add 2 tablespoons water and scrape up fond with wooden spoon. Continue to cook, stirring frequently, until fond has built up again, about 2 minutes longer. Add 2 more tablespoons water and scrape up browned bits. Repeat cooking, adding water, and scraping until onions are completely softened and a deep, dark brown, about 15 minutes total.
Add tomatoes, white sugar, brown sugar, lemon juice, vinegar, salt, and red pepper flakes in with the onions; stir to combine. Bring mixture to a boil, then reduce heat and simmer, stirring occasionally, until tomatoes have broken down and jam has thickened and developed a jammy consistency, 1 to 1 1/2 hours. Remove from heat. Transfer jam to a an airtight container and store in refrigerator for up to two weeks, or ladle into sterilized jars and process in a hot water bath for 10 minutes to seal for self storage.