How to Make It
Preheat the oven to 350°. In a medium saucepan, cook the butter over moderate heat, stirring, until fragrant and the solids turn dark golden, 5 to 7 minutes. Let cool.
Line a large rimmed baking sheet with parchment paper. In a stand mixer fitted with the paddle, beat the browned butter with the brown sugar until smooth. Beat in the flour and salt. Scrape the mixture onto the parchment paper–lined baking sheet and spread into a 9-by-12-inch rectangle. Bake for 12 to 15 minutes, until just set. Using a knife, score the shortbread into 12 squares. Let cool, then cut into squares.
In a medium saucepan, bring the cream just to a simmer over moderate heat. Remove from the heat and add the chocolate and a pinch of salt. Let stand for 2 minutes, then stir until the chocolate is melted. Let cool completely, then refrigerate until well chilled, about 30 minutes.
Pour the chocolate cream into a stand mixer fitted with the whisk. Beat at medium-high speed until thick and smooth; do not overbeat. Refrigerate the chocolate mousse for 30 minutes.
Reheat the oven to 350°. On a rimmed baking sheet, toss the strawberries with the sugar and a pinch of salt. Roast for about 10 minutes, until the strawberries are just softened and starting to release their juices. Transfer to a bowl and let cool completely. Serve the mousse with the shortbread and roasted strawberries.