Slice the goat cheese into twelve ½-inch-thick slices (a length of dental floss works well). Combine the egg white and water in a shallow dish and beat until frothy. Place the bread crumbs in another shallow dish. Dip each cheese slice in the egg white mixture; coat with the breadcrumbs, pressing to coat the cheese slices thoroughly.
Place the slices on a plate and chill for 30 minutes.
Whisk the strawberry jam, vinegar, salt, pepper, and ¼ cup olive oil in a small bowl. Pour over the arugula in a large salad bowl and toss well.
Divide among four plates and top with the strawberries and pecans.
Melt the butter with 1 tablespoon olive oil in a large skillet over medium-high heat. Add the goat cheese rounds. Cook for 1 minute on each side or until golden brown.
Top each salad with three warm rounds. Serve Arugula and Strawberry Salad immediately.
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