In a large bowl, beat together the softened butter and sugar until light and fluffy. Beat in egg and vanilla. Mix in the flour and salt until just incorporated. Form the dough into a disc, wrap in plastic, and refrigerate for 30 minutes.
Preheat oven to 350˚F (180˚C) and line two baking sheets with parchment or lightly grease.
Break off 1 inch (2.5 centimeter) pieces of dough and roll into a ball. Place on the prepared baking sheet 1 inch (2.5 centimeters) apart. Use your thumb or the back of a 1/2 teaspoon to slightly flatten the ball in the center, making an indention.
Bake in preheated oven until cooked through and barely golden on the bottom, 12-15 minutes. If the indentions are too shallow, lightly press to make them more pronounced. Remove to a wire rack and allow the cookies to cool completely.
Melt the white chocolate and heavy cream in a microwave safe bowl in 15 second increments, mixing between, just until melted. Be careful not to overheat.
Whisk to create a smooth and creamy mixture. Spoon or pipe the white chocolate into each of the cookie indentions. Let set, about 1 hour, before serving.