Beat the eggs, sugars, vanilla, and salt in a bowl with an electric mixer on medium-high until light and fluffy, 2 to 3 minutes. Add the peanut butter and beat until combined.
Shape the dough into balls (about 11/2 tablespoons each) and place 2 inches apart on the baking sheets. Gently make a well in the center of each cookie with your thumb or a 1/2-teaspoon measuring spoon.
Bake, rotating the sheets halfway through, until the edges are firm, 12 to 14 minutes.
While the cookies are still warm, use the measuring spoon to deepen the well in the center of each cookie. Let cool slightly on the baking sheets, then transfer to a wire rack to cool completely.
Bring the cream to a boil in a small pot. Remove from heat, add the chocolate, and let sit for 5 minutes. Whisk to combine. Let cool slightly, then spoon about 1 teaspoon into each well.
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