Tip the sun dried tomatoes into a small bowl and cover with 200ml boiling water. Set aside to soak for 25 mins while you make the rest of the stew.
Pour 1 tbsp olive oil into a large pan and bring to a medium-high heat. Slice the chicken fillets into chunky bite-sized pieces and add to the hot pan. Season well with salt and pepper and fry for 3-4 mins, turning once or twice, till golden brown. Transfer to a plate and set aside.
While the chicken cooks, peel and finely slice the onion and garlic. Add them both to the empty pan and reduce the heat to medium. Fry for 2-3 mins till softened.
Pour 750ml boiling water into the pan and crumble in the stock cube. Return the chicken to the pan and gently simmer for 5 mins.
Trim and peel the carrots and slice into thin batons. Slide into the pan and simmer for 6 mins.
Break the woody tips from the asparagus (gently bend them and they'll snap at the right point), and slice the tender spears in half. Shred the spring greens, discarding any thick cores. Fold the shredded greens into the stew, and cook for a further 3-4 mins or till all of the veg is tender.
Drain the sun dried tomatoes, reserving the soaking liquid, and tip them into a food processor. Pick the leaves from the tarragon and add to the tomatoes. Zest the lemon and scrape in, along with half of the juice, 1 tbsp olive oil and a pinch of salt and pepper. Blend well, adding a little of the tomato liquid to form a chunky pesto.
Taste the stew and season if necessary. Squeeze in the remaining lemon juice. Finely chop the parsley, leaves and stalks.
Ladle the stew into deep bowls and dollop on the tomato and tarragon pesto. Finish with the chopped parsley and a good crack of black pepper.