Preheat your oven to 350°F (175°C). Place four ramekins or small oven-safe dishes on a baking sheet for easy handling. lightly grease the ramekins with butter or nonstick spray to prevent sticking.
In a saucepan over medium heat, combine the heavy cream and whole milk. Heat until warm but not boiling, then remove from heat. In a mixing bowl, whisk the egg yolks until smooth. Gradually add a small amount of the warm cream mixture to the yolks, whisking constantly to temper the eggs (this prevents curdling). Return the tempered mixture to the saucepan and stir in the Parmesan cheese, salt, pepper, and paprika. Cook over low heat, stirring constantly, until the mixture thickens slightly (about 3-4 minutes). Remove from heat and let it cool slightly.
In a separate bowl, gently mix the lump crab meat with fresh chives, lemon juice, Dijon mustard, and Old Bay seasoning. Be careful not to break up the delicate crabmeat. Fold this mixture into the cooled brulee base, distributing the crab evenly.
Divide the crab brulee mixture evenly among the prepared ramekins. Smooth the tops with a spoon or spatula for an even surface. Place the baking sheet with the ramekins in the preheated oven and bake for 20-25 minutes, or until the brulees are set but still slightly jiggly in the center.
Once the brulees are baked, remove them from the oven and let them cool slightly. Sprinkle about 1/2 teaspoon of granulated sugar evenly over the top of each ramekin. Use a kitchen torch to caramelize the sugar until golden brown and bubbly. If you don’t have a torch, broil the brulees in the oven for 1-2 minutes, watching closely to avoid burning.