Combine chocolate chips, salt, and cayenne in a heat-proof measuring cup; set aside.

Separate eggs by cracking one egg into your hand over a bowl. Open your fingers slightly and gently jiggle your hand until the egg white falls into the bowl below. Transfer the yolk to a skillet. Repeat with remaining eggs. Reserve egg whites for another use.

Add sugar, milk, and cream to egg yolks. Whisk thoroughly, breaking egg yolks first, until well combined.

Place skillet on the stove over medium or medium-low heat. Cook, stirring constantly with a silicone spoon, until very hot and thick enough to coat the back of the spoon, about 5 minutes. An instant-read thermometer should read at least 175 degrees F (79 degrees C). Remove from the heat.

Set a fine sieve over the bowl of reserved chocolate. Strain the custard sauce into the chocolate and let sit for 2 minutes. Whisk until chocolate has melted and custard sauce is smooth and shiny, about 2 minutes. Add vanilla and butter; stir until butter has melted, about 1 minute.

Pour warm custard sauce into 6 serving glasses. Tilt each glass and rotate it around so the warm chocolate coats another 1/2 inch of the glass. Cover with plastic and place in the refrigerator until completely chilled, at least 3 to 4 hours.

Combine cream and vanilla extract for topping in a metal bowl and whisk until thickened; make sure no peaks form. Spoon cream into the glasses, then tilt and twirl to coat the sides a bit.

Garnish with shaved chocolate and serve.