• Peel the cucumber if desired and trim the ends. Lay the cucumber flat; using a peeler, shave the cucumber lengthwise into thin ribbons. 
• Trim the root ends from the scallions; cut the scallions on the diagonal into 2-inch lengths.

In a small sauce pot, bring the quick-pickle brine and 2 tablespoons [¼ cup] water to a boil. Remove from the heat, add the cucumber, scallions, and carrots, and season with salt and pepper. Cover and let stand, stirring occasionally, while you prepare the rest of the meal. 

Bring a medium sauce pot of water to a boil. Add the udon and cook, stirring occasionally to prevent sticking, until just tender, 4 to 5 minutes. Drain and rinse with cold water, then return them to the pot and toss with 1 to 2 teaspoons oil. Set aside. 

While the water is heating and the udon is cooking, prepare the tofu.

• Pat the tofu dry with paper towels and cut it into ½-inch cubes.
• Grate or peel and finely chop enough ginger to measure 2 teaspoons [4 tsp].
• Cut the tomatoes in half. Season generously with salt and pepper. Set aside for serving.

In a large frying pan over medium-high heat, warm 1 to 2 tablespoons oil until hot but not smoking. Add the tofu and ginger and cook, stirring occasionally, until the tofu is browned and crisp all over, 3 to 4 minutes. Stir in the tamari and cook until thickened slightly and the tofu is coated, 1 to 2 minutes. Remove from the heat and season to taste with salt and pepper. 

Transfer the udon to individual bowls. Using a slotted spoon, remove the pickled vegetables from the brine. Top the udon with the pickled vegetables, tofu, and tomatoes. Garnish with the sesame seeds and serve.