Prepare the Hazelnut Financiers: Cook butter in a small saucepan over low, stirring occasionally, until it just begins to brown, about 10 minutes. (It can burn quickly, so remove pan from heat when foam subsides and browning process begins.) Cool to room temperature, about 30 minutes.

Whisk together powdered sugar, hazelnut meal, almond flour, all-purpose flour, baking powder, and salt in a medium bowl. Add egg whites, 1 at a time, whisking just to combine. (Do not overwork batter.) Gradually whisk butter into flour mixture. Cover with plastic wrap; chill 3 hours.

Meanwhile, prepare the Strawberry Jam: Bring strawberries, cane sugar, and lemon juice to a boil in a medium saucepan over medium-high, stirring occasionally. Reduce heat to medium-low, and simmer, stirring often, until strawberry mixture thickens, about 30 minutes. Mash berry mixture with a potato masher to break up fruit. Transfer jam to a small bowl; cool to room temperature, about 1 hour.

Prepare the Lemon Puree: Bring granulated sugar and 3 cups water to a boil in a small saucepan over high. Reduce heat to medium-low; add whole lemons. Simmer until lemons are completely tender, 1 hour. Remove lemons, reserving cooking liquid. Cool lemons and liquid 10 minutes.

Place lemons and 1 cup of the reserved cooking liquid in a blender; process on high until mixture is the consistency of applesauce, about 1 minute, adding more reserved cooking liquid as needed. Pour lemon mixture through a fine wire-mesh strainer into a medium bowl, discarding solids. Stir salt into lemon mixture.

Preheat oven to 375°F. Coat a standard 12-cup muffin pan with baking spray. Spoon chilled batter into a ziplock plastic freezer bag; snip a 1⁄2-inch hole in 1 corner. Pipe batter into cavities of pan, filling each halfway full. Bake in preheated oven until golden brown, 16 to 18 minutes. Cool in pan on a wire rack, about 30 minutes.

Top financiers with powdered sugar and sliced strawberries. Serve with Strawberry Jam and Lemon Puree.