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Lay out brioche slices. Add 2 tablespoons jam to the center of 2, then top with remaining slices. Press edges lightly to create a seal.
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Fill a deep, wide skillet with 2-inches oil and set over medium-high/high heat until oil reaches 340 degrees F. (Or use a countertop electric skillet.)
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While oil heats, whisk to combine eggs, flour, milk, and sea salt. Spread turbinado or caster sugar out in a shallow dish and set aside.
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Working quickly, dip sandwich in batter, carefully lift out, letting some of the batter drain off, lower into the hot oil, and fry each side until light golden – 30 seconds to 1 minute per side. Use a slotted fish spatula to lift donut out of oil, letting any excess oil drip off, then set in sugar filled dish and turn to coat. Repeat for second sandwich.
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Slice sandwiches in half, and enjoy immediately (watch out for the hot jam!).