Pat chicken breasts dry with paper towels. Season evenly with ½ tablespoon kosher salt and ½ teaspoon black pepper. Allow to rest at room temperature for 15 minutes.
In a large pot, combine chicken stock, remaining salt and pepper, bay leaves, thyme, lemon slices, onion, and garlic. Bring to a gentle simmer over medium heat.
Reduce heat to low once simmering. The ideal poaching temperature is 165°F (74°C). Use your digital thermometer to verify.
Gently lower chicken breasts into the barely simmering liquid. The chicken should be fully submerged; add water if necessary.
Cover and maintain the gentle simmer. Avoid boiling. Poach until internal temperature reaches 165°F (74°C), approximately 15-18 minutes depending on thickness.
Remove pot from heat. Leave chicken in the liquid for 5-10 minutes. This resting period allows for moisture redistribution within the protein fibers.
Remove chicken using tongs and transfer to a clean cutting board. Strain 1 cup of the poaching liquid through a fine mesh strainer and reserve.
Allow chicken to cool slightly (about 5 minutes) before shredding. Using two forks or meat claws, shred the chicken with the grain for longer strands or against the grain for shorter pieces.
Toss the shredded chicken with 2-4 tablespoons of the strained poaching liquid to restore moisture and enhance flavor.
Divide into meal prep containers. Drizzle each portion with a small amount of olive oil to create a moisture barrier. Cool completely before refrigerating.