How to Make It
Fill a large metal bowl with ice water and set another large metal bowl in it. Set a strainer in the bowl.
In a medium bowl, whisk the egg yolks with 1/4 cup of the sugar until slightly pale. In a medium saucepan, heat the milk with the cream and vanilla beans and seeds until small bubbles appear around the edge. Add the hot milk to the egg mixture in a thin stream, whisking constantly. Return the vanilla mixture to the saucepan and cook over moderate heat, stirring constantly with a wooden spoon until slightly thickened, about 5 minutes. Strain the vanilla sauce into the bowl in the ice bath and let cool completely, stirring occasionally.
Meanwhile, in a saucepan, combine the strawberries, orange juice, zest, vinegar and the remaining 1/4 cup of sugar; season very lightly with salt and pepper. Bring to a boil and cook over moderately high heat, stirring frequently, until the mixture has become jammy, about 10 minutes. Transfer the jam to a bowl and let cool.
Heat a panini press or griddle over low heat. Spread the mascarpone on 4 of the brioche slices. Spoon 3 tablespoons of the jam on each of the other 4 slices and close the sandwiches. Brush the sandwiches with the butter and grill or press until golden, about 5 minutes. Transfer the panini to a cutting board and dust with confectioners' sugar. Halve each sandwich.
Put 1 pint of ice cream and half of the vanilla sauce in a blender and blend until smooth. Pour into 2 large glasses. Repeat with the remaining ice cream and vanilla sauce to make 2 more shakes. Serve the milk shakes right away, with the panini.