Put half-and-half, lemon zest, and cinnamon stick into a heavy medium pot and bring to a simmer over medium heat. Strain into a large bowl, discarding solids, and set aside, stirring often with a wooden spoon, to let cool completely.
Whisk egg yolks and 1⁄2 cup of the sugar together in a large mixing bowl until thick and pale yellow, then whisk in cooled infused half-and-half. Transfer to a heavy medium pot and cook over medium-low heat, stirring constantly with the wooden spoon, until thick enough to coat the back of the spoon, 25-30 minutes. Strain custard into a large bowl and set aside, stirring frequently, to let cool completely. Cover and refrigerate until well chilled.
Meanwhile, grease a medium sheet pan with butter and set aside. Put remaining sugar and 4 tbsp. warm water into a small heavy pot and simmer over medium-low heat, swirling pot over heat frequently, until sugar caramelizes, about 25 minutes. Immediately pour syrup onto prepared sheet pan and tilt pan around, spreading syrup out to a thin layer. Set aside to let cool. Crack caramel into small pieces and stir into cooled custard . Process in an ice cream maker according to a manufacturer's directions. Serve with carquinyolis, if you like.