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1
Using a paring knife or a vegetable peeler to remove the zest from the lemons in large strips, then reserve them for later. Juice the lemons, you should get around 1/4 cup.
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2
Make a lemon syrup: Combine the lemon juice and the 1 tablespoon of sugar in a small saucepan and simmer over medium heat until the sugar is dissolved. Remove from the heat and refrigerate.
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3
Shake the cans of coconut milk well before opening. Set aside about 1/2 cup of coconut milk. Combine the remaining coconut milk, remaining 1/2 cup sugar, and salt in a saucepan. Cook at medium heat until all of the ingredients combine smoothly and the mixture is uniformly warmed, about 2 minutes. (It's okay if it comes to a simmer, but it might not.)
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4
In a small bowl, whisk the reserved 1/2 cup coconut milk with the cornstarch until well combined. Stir the cornstarch slurry and the reserved lemon zest strips into the warm coconut milk base. Cook over medium heat until the mixture has thickened enough to coat the back of a spoon, 5 to 7 minutes. (Again, it's okay if the mixture simmers, but do not to allow the mixture to boil.)
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5
Remove from the heat and pour the mixture into a separate bowl. Allow the mixture to cool on the counter for 30 minutes, then cover with plastic wrap and chill in the refrigerator for 4 to 5 hours, until very cold.
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6
While the base is cooling, make the violet-mulberry sauce: Add mulberries and sugar to a medium saucepan. Simmer over medium heat and cook for about 5 minutes, or until the berries start to lose their shape and become juicy. Remove from the heat, and allow to cool slightly. Once slightly cool, add the violet liqueur, stir to combine, and blend with an immersion blender (or in a food processor or blender), just until all of the berries break down, and a thick sauce forms. Press through a fine-mesh strainer to remove the mulberry stems and some of the seeds. (Some of the seeds will remain, I don't mind them, but if you do, use a really-fine-mesh strainer and/or line the strainer with cheesecloth). Set sauce aside.
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7
Once coconut milk base is very cold, remove the strips of lemon zest. Pour it into your frozen ice cream canister and turn on the machine. Pour the lemon syrup through the opening in the top of the machine and continue to process until the ice cream is thick and creamy.
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8
Pack the ice cream into a storage container, alternating ice cream and violet-mulberry sauce, creating pockets of sauce. Press a sheet of parchment paper directly against the surface and seal the container with an airtight lid. Freeze until firm, at least 4 hours.
