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1. Bring butter, granulated sugar, brown sugar, water, corn syrup, lemon juice, and salt to boil in large saucepan over medium heat, stirring occasionally to dissolve sugar and melt butter. Once mixture is at full rolling boil, cook, stirring occasionally, for 5 minutes (caramel will register about 240 degrees). Immediately reduce heat to medium-low and gently simmer (caramel should maintain steady stream of lazy bubbles-if not, adjust heat accordingly), stirring frequently, until mixture is color of dark peanut butter, 12 to 16 minutes longer (caramel will register about 300 degrees and should have slight burnt smell).
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2. Remove pan from heat; carefully pour 1/4 cup cream into caramel mixture and swirl to incorporate (mixture will bubble and steam); let bubbling subside. Whisk vigorously and scrape corners of pan until mixture is completely smooth, at least 30 seconds. Return pan to medium heat and gradually whisk in remaining 3/4 cup cream until smooth. Whisk in 2 cups milk until mixture is smooth, making sure to scrape corners and edges of pan to remove any remaining bits of caramel.
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3. Meanwhile, microwave remaining 1/4 cup milk until simmering, 30 to 45 seconds. Whisk egg yolks and cornstarch together in large bowl until smooth. Gradually whisk in hot milk until smooth; set aside (don't refrigerate).
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4. Return saucepan to medium-high heat and bring mixture to full rolling boil, whisking frequently. Once mixture is boiling rapidly and beginning to climb toward top of pan, immediately pour into bowl with yolk mixture in 1 motion (do not add gradually). Whisk thoroughly for 10 to 15 seconds (mixture will thicken after a few seconds). Whisk in vanilla and rum. Spray piece of parchment paper with vegetable oil spray and press on surface of pudding. Refrigerate until cold and set, at least 3 hours. Whisk pudding until smooth before serving.
