Make the compote: Combine the strawberries, rhubarb, 1/3 cup sugar, the vanilla and cloves in a medium saucepan over medium-high heat; cook, stirring occasionally, until broken down and thickened, 20 to 25 minutes. Remove from the heat and let cool to room temperature. (You can make the compote up to 2 days ahead; cover and refrigerate.)
Cut a 3-inch-wide slit into one side of each bread slice, cutting about three-quarters of the way through to make a pocket. Fill with the compote, reserving about 1/4 cup for topping.
Whisk the eggs, milk and the remaining 2 tablespoons sugar in a large baking dish. Add the stuffed bread and soak, about 5 minutes per side.
Melt the butter in a large nonstick skillet over medium heat. Add the stuffed bread and cook until golden brown and cooked through, 5 to 6 minutes per side, adjusting the heat if necessary. Top with the reserved compote and yogurt.