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1
Combine the rhubarb and 1/2 cup sugar in a saucepan; cook over medium heat, stirring occasionally, until the rhubarb is tender, and dissolves into a loose puree as you stir; remove from heat and stir in the raspberry jam; let cool completely.
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2
In a big bowl, combine the half-and-half, eggs, vanilla, salt, and remaining 1/4 cup sugar; whisk together until smooth.
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3
Pour the batter into a large shallow dish and set aside.
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4
Position a rack in the middle of the oven and preheat to 350.
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5
Slice the brioche crosswise into six 2-inch thick slices; with a small, sharp knife, cut a deep pocket into each piece of brioche, starting at the top of the slice--as if you are going to cut the thick piece into two 1-inch thick slices, opening the bread up, but leaving the sides connected and not cutting all the way through to the bottom.
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6
Gently open each pocket, taking care not to rip the bread, and spoon 2-3 tablespoons of the rhubarb mixture into each slice of brioche.
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7
Dip 2 or 3 sandwiches into the batter and let soak for about 1 minute per side.
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8
Melt 2 tablespoons of butter and 2 tablespoons oil in a large nonstick skillet over medium heat.
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9
When the fats are melted and start to bubble, lift sandwiches from the batter, allowing any excess to drip off, coat each side in the panko crumbs and place in the hot skillet; cook until golden brown, 2-3 minutes per side.
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10
Place each sandwich on an ungreased baking sheet; wipe out the skillet and melt the remaining butter and oil over medium heat; cook the remaining 3 sandwiches as you did the first batch.
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11
Place all 6 sandwiches on the ungreased baking sheet and bake for 5-6 minutes; serve warm, sliced on the diagonal, sprinkled with powdered sugar, and accompanied by whipped cream or ice cream, if desired.
