Special equipment: a silicone baking mat
For the French toast: At least 2 hours and up to 1 day ahead, preheat the oven to 350 degrees F. Place the bread slices on a rack on a baking sheet. Put the bread in the preheated oven, then immediately turn the oven off and let the bread sit until it begins to dry out, 20 to 30 minutes.
Whisk together the eggs, yolks and granulated sugar until smooth. Whisk in the half-and-half, orange liqueur, vanilla, orange zest and salt. Lay the bread on a rimmed baking sheet or in a large baking dish, pour the custard mixture over the top and let sit for 15 minutes. Flip and let sit for at least 30 minutes more. (You can also cover and refrigerate overnight, turning a few more times.)
For the strawberries: Meanwhile, combine the strawberries, granulated sugar and orange liqueur in a bowl. Let sit at room temperature until the strawberries release their juices and soften slightly, about 30 minutes.
Preheat the oven to 375 degrees F. While the bread is soaking, melt the butter in a small saucepan over high heat, then add the muscovado sugar and whisk until smooth. Transfer to a baking sheet lined with a silicone baking mat and spread the mixture evenly over it. Place the soaked bread directly on the sugar. Bake until golden brown and puffed, about 20 minutes.
Serve the French toast, sugar-side up, topped with the berries and their juices.