Fill a medium saucepan halfway with water and bring to a rolling boil while you trim the green beans and cut them in half.
Prepare the hard-boiled eggs using your preferred method; cool in ice water, peel, and cut each egg lengthwise into quarters.
When the water is boiling, add the green beans and cook exactly 3 minutes, then drain and immediately transfer the beans to an ice water bath to stop cooking.
Drain the beans thoroughly in a colander and set aside.
Cut the tomatoes into quarters (or smaller pieces if large) and halve the Kalamata olives; measure the drained capers.
Trim the radishes and cut each lengthwise into quarters.
Open the tuna cans, drain and discard the oil, then break the tuna into slightly chunky pieces with a fork and transfer to a bowl or serving platter.
Whisk together caper brine, lemon juice, olive oil, Dijon mustard, minced garlic, Spike Seasoning (if using), and dried thyme until emulsified.
Stir the drained capers into the dressing, then season with salt and freshly ground pepper to taste.
Arrange the green beans, quartered eggs, tomatoes, olives, radishes, capers, and tuna in sections on a large serving platter.
Whisk the dressing again and drizzle some over the salad, making sure to distribute capers across the platter; reserve remaining dressing to pass at the table.
Serve the composed Nicoise Tuna Salad immediately.