1. Season tuna with salt. Stir 3 cups oil, the lemon zest, garlic, and thyme in a medium saucepan. Add tuna and more oil to cover tuna by 1/2 inch if needed. Set over medium-low heat. Cook, covered, until tuna is opaque, 30 to 40 minutes. Remove from heat; let cool completely. Tuna in oil can be refrigerated in an airtight container up to 3 days.

2. To serve, remove tuna from cooking oil with tongs; transfer to paper towels to drain. Discard cooking oil. Flake into 2-inch pieces. Drizzle with oil; sprinkle with pepper. Serve with lemon and the tomato-olive salad. Garnish with thyme.