Whisk together eggs, sugar and salt in a large bowl. Add the milk, cream and peach puree and vanilla and whisk until combined. Place brioche in a large baking dish, pour custard over and turn to coat. Let sit for 5 minutes.
Heat 2 tablespoons of butter and 2 tablespoons of oil in a large nonstick saute pan over high heat. Add 4 slices of the soaked bread and cook until golden brown, flip over and continue cooking until golden brown on the second side. Place 1 slice of the French toast on a plate, ladle some of the peach syrup on top. Top with a second slice of toast and ladle more of the peaches on top. Garnish with a mint sprig.
White Peach-Pecan Maple Syrup:
Heat maple syrup over low heat in a medium saucepan. Add the peaches, pecans and bourbon and let simmer for 10 minutes.