Coat a 9 x 13-inch baking dish with cooking spray. In a medium bowl, whisk together eggs, milk, dry mustard, salt, and a grind of pepper.
Arrange half the bread cubes in baking dish in a single layer. Pour half the egg mixture over bread. Top with half the thyme, ham, and cheese. Add remaining bread to egg mixture in bowl; toss well. Arrange bread on top of first layer in baking dish. Pour any remaining egg mixture over top. Top with remaining thyme, ham, and cheese.
Cover and refrigerate at least 4 hours, or overnight. Let sit at room temperature 30 minutes before baking.
Heat oven to 350°F. Uncover strata. Bake until set and lightly golden, about 45 minutes. Let rest 10 minutes before serving. Dust top with sugar; serve with jelly on the side.