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1. In a medium heatproof bowl, whisk the yolks just to break them up, then whisk in half of the brown sugar (1/4 cup). Set aside.
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2. In a heavy nonreactive saucepan, stir together the cream, milk, salt, and the remaining brown sugar (1/4 cup) and put the pan over medium-high heat. When the mixture approaches a bare simmer, reduce the heat to medium.
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3. Carefully scoop out about 1/2 cup of the hot cream mixture and, whisking the eggs constantly, add the cream to the bowl with the egg yolks. Repeat, adding another 1/2 cup of the hot cream to the bowl with the yolks. Using a heatproof rubber spatula, stir the cream in the saucepan as you slowly pour the egg-and-cream mixture from the bowl into the pan.
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4. Cook the mixture carefully over medium heat, stirring constantly, until it is thickened, coats the back of a spatula, and holds a clear path when you run your finger across the spatula, 1 to 2 minutes longer.
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5. Strain the base through a fine-mesh strainer into a clean container. Set the container into an ice-water bath, wash your spatula, and use it to stir the base occasionally until it is cool. Remove the container from the ice-water bath, cover with plastic wrap, and refrigerate the base for at least 2 hours or overnight.
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6. Set the cream by the stove so it's at hand when you need it. Put 2 tablespoons of the sugar in a heavy nonreactive saucepan, and put the pan over medium-high heat. When the sugar is melted around the edges and starts turn amber in places (about 2 minutes), stir the mixture gently and add another 2 tablespoons sugar to the pan.
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7. Continue to add the remaining sugar 2 tablespoons at a time, stirring frequently and allowing most of the sugar to melt before you add more. Watch carefully as the sugar darkens, stirring gently to help it melt evenly.
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8. When the caramel becomes a dark mahogany color, remove the pan from the heat and immediately but slowly pour the cream into the pan. (The mixture will steam and bubble up, so wear oven mitts and be very careful to avoid splatters and steam burns.) When the bubbling subsides, gently stir to completely blend the cream into the caramel. If you have lumps of hardened caramel in your pan, simply put the pan over low heat and stir until the caramel is melted.
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9. Stir in the ginger paste and salt and let cool. (You can make the swirl up to 2 weeks ahead of time. Keep it refrigerated, but let it come to room temperature before using.)
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10. When the base is completely chilled, freeze in your ice cream machine according to the manufacturer's instructions. While the ice cream is churning, put the container you'll use to store the ice cream into the freezer.
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11. As you transfer the ice cream to the storage container, drizzle in the caramel after every few scoopfuls, using about 6 tablespoons of the caramel (or more if you like). When all the ice cream is in the container, use a chopstick or butter knife to gently swirl the mixture. Enjoy right away or, for a firmer ice cream, freeze for at least 4 hours.
