Preheat the oven to 450F or 425F convection. Line a rimmed baking sheet with parchment paper.
Add the potatoes, olive oil, ½ teaspoon salt, and black pepper to the baking sheet and toss to coat. Bake for 20-25 minutes, or until crispy and golden.
Place the chicken breasts in a large pot. Add enough water to just barely cover the chicken (about 3 cups) along with a generous amount of salt to season the chicken (about 2 tablespoons, you won’t be eating all that salt don’t worry!). Bring to a rapid boil, then reduce the heat to low. Flip each breast over, and simmer, covered, very gently for 10-15 minutes or until cooked through (160F), depending on how thick the chicken is. Transfer chicken to a cutting board, let it rest for 5 minutes, then slice it on a diagonal.
To make the romesco, add all ingredients to a high-speed blender and blend until smooth. To make it thicker, add more toasted nuts. Taste and add more salt or lemon, if needed.
Place the fennel in a large bowl and toss with 1 tablespoon lemon juice and remaining ¼ teaspoon salt.
Serve the warm chicken and potatoes smothered with romesco sauce and a side of crunchy fennel. Top with parsley, if desired.