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Set a fine-mesh sieve over a medium bowl and place it near your stove. Place the chopped chocolate in a second medium bowl nearby.
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In a large saucepan, whisk together the cornstarch, sugar, and salt. Add the egg yolks and whisk until fully combined, making sure to reach into the edges of the pan.
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Pour in 2 1/2 cups (568g) milk and the heavy cream, then whisk thoroughly to eliminate as many lumps as possible.
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Set the saucepan over medium-low heat and cook, whisking frequently, for 8 to 10 minutes, until the mixture is thick enough that your whisk leaves a trail. Stir constantly toward the end, switching to a rubber spatula if necessary to reach the edges of the pan.
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Remove the saucepan from the heat and whisk in the sweetened condensed milk until the pudding is smooth and shiny. Immediately press the pudding through the sieve to remove any lumps.
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Transfer about half of the pudding (about 1 1/2 cups/500g) to the bowl with the chopped chocolate and whisk until the chocolate is melted and the pudding is smooth. Add the remaining 2 teaspoons milk and whisk to combine.
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Whisk the vanilla bean paste into the plain pudding.
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Distribute the chocolate pudding, followed by the vanilla pudding, evenly among small cups, jars, or serving dishes (see “tips,” below for portioning guidance). Press a piece of plastic wrap or parchment paper directly onto the surface of each pudding to prevent a skin from forming, then cover and chill until set, about 3 hours.
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Store the pudding cups in the refrigerator for up to 5 days.
