Press or squeeze ginger using a fine sieve or cheesecloth until enough juice is extracted to measure 1 tablespoon.
Heat a well-seasoned large ridged grill pan over high heat until hot. Pat chicken dry and season with salt and pepper. Mix together soy sauce, mirin, ginger juice, chile paste, and sesame oil. Set aside 1/2 cup of soy mixture and toss remainder with chicken.
Grill chicken until just cooked through, 4 to 5 minutes on each side. When cool enough to handle, tear into bite-size pieces.
If using dried noodles, cook in a large pot of boiling water until al dente, about 12 minutes. Stir in spinach during last minute of cooking. (If using fresh udon, add noodles and spinach simultaneously to boiling water and cook 10 to 20 seconds total.) Drain noodles and spinach and transfer to a bowl of ice and cold water. When cold, drain well in a colander.
Toss noodles and spinach with reserved soy mixture in a large bowl.
Serve noodles and spinach topped with chicken.
Chicken can be grilled on an outdoor grill - lightly oil rack and set 5 to 6 inches over glowing coals.