1 cup, plus 8 tablespoons, unsalted butter, divided
¼ cup sunflower seeds, roasted, for garnish
1 teaspoon flake salt, for garnish
Make the ice cream: In a medium bowl, whisk together the sunflower seed butter, egg yolks and whole egg, and set aside. In a medium saucepan, whisk the milk, cream, light corn syrup, honey and salt, and bring to a simmer over medium heat. Remove the pan from the heat and pour ¼ cup of the hot liquid mixture into the bowl with the egg mixture. Whisk until all the liquid has been fully incorporated; repeat this step until there's about a third of the liquid left. Add the remaining third at once and whisk together.
Pour the mixture back into the saucepan and, while constantly whisking, cook over medium heat until the ice cream base is formed and coats the back of a spoon, 5 to 8 minutes. Pour the mixture through a fine-mesh strainer into a bowl over ice water, cover with plastic wrap directly and allow the mixture to completely cool.
Churn the ice cream in an ice cream maker. Once frozen, spread the ice cream into a parchment-lined 9-by-11-inch baking pan. Freeze until the ice cream is solid, 2 hours.
Make the sunflower seed caramel: In a large sauté pan, heat the sugar over medium heat. Stirring frequently, let the sugar fully melt and cook to a dark, golden-brown caramel, 5 to 7 minutes. Whisk in the butter one cube at a time until each cube is fully melted and incorporated into the sugar. Add the heavy cream and mix fully. Remove the sauté pan from the heat, add the salt and sunflower seeds, then let completely cool.
Once fully cool, pour the caramel over the frozen ice cream and spread evenly. Freeze overnight.
Remove the ice cream from the pan and cut into 1-by-4-inch rectangles. Store the bars on a parchment-lined baking sheet and place back in the freezer to fully firm up, 1 hour.
Meanwhile, make the chocolate shell: In a heated double boiler, heat 1 cup of the semisweet chocolate chips and 6 tablespoons of the unsalted butter, and mix together until the mixture is fully melted and combined, 5 minutes.
Using an offset spatula, slather a small amount of chocolate on the ice cream side of each bar. Place the bars chocolate-sides down back onto the parchment-lined baking sheet and place in the freezer to firm back up, 1 hour.
While the bottom chocolate layers of the bars firm up, make the rest of the chocolate shell: Add the remaining 3 cups of semisweet chocolate chips and 1 cup and 2 tablespoons of unsalted butter to a double boiler. Once the mixture is fully melted, 8 to 10 minutes, lower the heat and keep the chocolate warm.
Move the ice cream bars, chocolate-sides down, to a wire rack with a baking sheet underneath. Spoon the melted chocolate over the bars until the bars are fully coated with chocolate on all sides. While the chocolate is still soft, sprinkle the tops of the bars with toasted sunflower seeds and a pinch of flake salt. Place the bars back in the freezer to firm up, 2 hours. Serve immediately out of the freezer.