Bake the patties for 10 minutes. Remove the sheet from the oven and lay a small piece of pepper jack cheese over each burger. Return them to the oven for another 1 to 2 minutes, just until the cheese is melted and bubbling, and the burgers are cooked through.
While the burgers are in the oven, you can spread a little extra mustard on the hamburger buns if you like, then cut the buns into 1-inch pieces. Have your baby spinach and grape tomatoes ready for assembly.
To assemble, thread a toothpick through a piece of bun (mustard side down, if using), then a baby spinach leaf, then a grape tomato, then a burger, and finally the top piece of bun (mustard side up). When arranging them on a serving dish, press each assembled burger down slightly; this helps them stand upright.