Slice tomatoes in half and dehydrate at 170? for 4 to 5 hours.
Bring stock to a boil. Combine and mix dry ingredients, then pour stock in the center of the dry ingredients. Add beaten egg and mix well. The texture of the dough should be firm but not dry.
Roll dough out on a well-floured surface until about 1/8 of an inch thick, then cut into strips ? inch wide.
Cook noodles in salted boiling water until al dente, 5 to 7 minutes.
Toss carrot bottoms and garlic in olive oil, then roast at 450 degrees convection for 20 minutes. Blend arugula, lemon juice, roasted carrot, garlic, and pine nuts until you have achieved a rough blended texture.
Mix in cheese and continue to blend. Adjust with salt and more lemon juice if needed.
Preheat grill to medium-high heat (between 400-450° Fahrenheit).
Place chicken thighs on grill grates and grill for approximately 6-7 minutes per side, or until the internal temperature in the thickest part of the thigh reads 165° Fahrenheit.
Remove chicken thighs to a plate or serving tray, cover lightly with foil, and allow them to rest for 10 minutes before serving.
In a saucepan, cover dehydrated tomatoes with water and bring to a boil, then simmer until tomatoes are soft and you have reduced water to ? cup.
Add ? cup of cold butter one piece at a time and whisk so butter is suspended in tomato water.
Mix noodles with tomato and butter, then top with a dollop of pesto and a grilled chicken thigh.