Bring a large pot of salted water to a boil. Pat the shrimp dry with paper towels. Cut the lemon in half, juice 1 half, and cut the other half into 4 quarters. Cut the broccoli into small florets (about 3/4-inch in size).
Combine the lemon juice with the chicken broth, soy sauce, and cornstarch.
When the water comes to a boil, add the udon noodles to the pot and cook stirring occasionally for about 6 minutes until the noodles are just tender (check package instructions as cooking times vary). Drain, rinse briefly with cold water to stop the cooking, drain again, then transfer to a shallow serving bowl or platter, and toss with the sesame oil.
Heat a wok or a very large skillet over medium high heat until the pan is hot. Add the vegetable oil, then immediately add the broccoli florets along with the garlic and ginger. Cook, stirring frequently, for 2 minutes (If the garlic and ginger are starting to burn, add a couple of tablespoons of water. Add the shrimp and continue to cook, stirring frequently, until the broccoli is crisp-tender and the shrimp have turned mostly pink, about 2 minutes. Stir in the scallions.
Add the sauce to the pan, and stir frequently until the sauce has thickened and the shrimp are completely cooked, about 2 minutes more. Turn the noodles back into the pan and use tongs to toss and combine the noodles, shrimp and broccoli with the sauce.
Serve the stir fry hot, with the lemon pieces on the side for people to squeeze over as they wish.