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. You can make the compote up to 2 days ahead; cover and refrigerate.
Whisk the eggs, milk and the remaining 2 tablespoons sugar in a large baking dish. Add the french bread and soak, about 5 minutes each side.
Melt the butter in a large skillet over medium heat. (You can also use a pancake griddle.) Add the bread and cook until golden brown and cooked through, about 5 minutes per side. Top with the compote and drizzle with yogurt.