Heat cream, milk, and sugar over medium heat, stirring frequently, until sugar is dissolved, about 5-7 minutes. Remove from heat and stir in vanilla. Cool completely. Chill well in the fridge or in an ice bath in a sealed container.
1 1/2 cups heavy cream, 1 1/2 cups whole milk, 1 3/4 cups granulated sugar, 1 teaspoon vanilla extract
Mean while, blend raspberries and 2 remaining tablespoons of sugar until smooth, then press through a fine mesh sieve to strain the seeds.
8 ounces raspberries
When the sherbet base is thoroughly chilled, churn in an ice cream maker.
Transfer 2/3 of the sherbet base to a bowl and stick it in the freezer. Add the orange juice and orange zest, if using, as well as a drop of orange food coloring to the remaining 1/3 of the sherbet base and churn to combine until about the consistency of soft-serve. This should only take a couple of additional minutes. Transfer to a bowl and stick in the freezer.
3/4 cup orange juice, 1 Tablespoon orange zest, Drop of orange food coloring
Add 1/3 of the plain sherbet base back into the mixer (it's okay if some of the orange flavor is still present - they are all getting mixed together anyway in the end). Add lime juice and lime zest, if using, as well as a drop of green food coloring. Churn until the consistency of soft-serve, another 2-3 minutes. Transfer to a bowl and stick in the freezer.
1/2 cup lime juice, 1 Tablespoon lime zest, Drop of green food coloring
Return final 1/3 of the plain sherbet base to the mixer bowl along with the raspberry puree and churn until the consistency of soft serve.
Take turns adding spoonfuls of each of flavor to a bread pan or other freezer-safe container, mixing them slightly, then transfer to the freezer for 4-6 hours until hardened.