Cut the lettuce into large pieces and place in a deep salad bowl. Put the whole cherry tomatoes in with the lettuce and toss.
Add the grilled pieces of chicken to the salad while still warm, together with the grated parmesan and mix. For the dressing, first clean the dill, chop finely and crush the garlic. In a large bowl, mix the dill, garlic, mayonnaise, mustard and olive oil, then add lemon juice and salt and beat well. Drizzle over the salad. Arrange cheese-flavored Crax sticks vertically in the salad, spaced evenly, and serve. Bon appetite!