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1
In a large saucepan, combine butter, both sugars, water, corn syrup, lemon juice and salt. Bring to a boil over medium heat, stirring occasionally until sugar is dissolved and butter is melted. Bring to a full rolling boil and cook, stirring occasionally, for 5 minutes (caramel will be about 240F). Immediately reduce to medium-low, until caramel develops a steady stream of lazy bubbles. Cook, stirring frequently, until mixture is color of dark peanut butter, smells ever-so-slightly burnt and registers 300F, 12-16 minutes more.
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2
Remove from heat and carefully add 1/4 c cream (it will bubble and steam), swirling to incorporate. When bubbles subside, whisk vigorously, scraping into corners, until completely smooth, at least 30 seconds. Over medium heat, whisk in remaining cream until smooth. Whisk in 2 c milk until smooth, remembering to scrape into the corners of the pan to collect any remaining caramel.
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3
In microwave, bring remaining 1/4 c milk to a simmer, 30-45 seconds. Whisk egg yolks and cornstarch together in a large heat-safe bowl until smooth. Gradually whisk in hot milk until smooth; set aside at room temperature.
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4
Return saucepan to medium-high heat and bring back to a full rolling boil, whisking frequently. When bubbles begin to climb sides of pan, immediately dump entire pan into bowl with yolk mixture (do not add slowly). Whisk vigorously and thoroughly for 10-15 seconds. Add vanilla and rum and whisk to combine. Spray a disc of parchment paper cut to fit the bowl with non-stick spray and press onto surface of pudding. Refrigerate until cold and set, 3 hours or more. Whisk until smooth before serving.
