Preheat an oven to 350 degrees F (175 degrees C). Grease a 9x13 inch baking dish.
Fill a large pot with lightly salted water and bring to a rolling boil over high heat. Once the water is boiling, stir in the lasagna, and return to a boil. Cook the pasta uncovered, stirring occasionally, until the pasta has cooked through, but is still firm to the bite, 8 to 9 minutes. Drain well in a colander set in the sink.
Melt the butter in a saucepan over medium-low heat. Whisk in the flour, and stir until the mixture becomes paste-like and light golden brown, about 5 minutes. Gradually whisk the chicken broth and 1/2 of the milk, into the flour mixture, and bring to a simmer over medium heat. Cook and stir until the mixture is thick and smooth, 10 to 15 minutes. Stir in the remaining milk and 1/2 of the minced garlic.
Place soda crackers in a resealable plastic bag. Finely crush the crackers, then add the Italian seasoning. Combine tuna, mixed vegetables, egg white, salt, 1/2 cup of Cheddar cheese, 1/4 cup of the cracker crumbs, 1/2 cup of the flour mixture, and the remaining minced garlic in a large bowl.
Spread a thin layer of the white sauce onto the prepared baking dish, followed by a layer of lasagna noodles. Spread about 1/3 of the tuna mixture over the noodles. Repeat the noodle and tuna layering three more times, topping with the remaining flour mixture. Sprinkle the pepper evenly on top of the casserole. Cover with aluminum foil.
Bake in the preheated oven for 35 minutes. Remove from the oven and top with 1/2 cup of Cheddar cheese and the remaining cracker crumbs. Change the oven setting to broil and return the casserole to the oven. Broil until lightly brown, 2 to 3 minutes.