Preheat your oven to 400°F (200°C). Line two baking sheets with parchment paper. On one baking sheet, add the Sweet Petites carrots and toss them with oil. Spread them in a single layer and bake for 15 minutes. Remove and let cool.
In a food processor, whip the goat cheese, cream cheese, and heavy cream on high until smooth and airy, about 30 seconds. Add more heavy cream if needed for a spreadable texture. Season with kosher salt to taste.
In a small bowl, combine the brown sugar, cinnamon, nutmeg, allspice, ground cloves, and ground ginger with a pinch of kosher salt. Melt the butter in a skillet over medium heat, then add the pecans and sugar mixture. Stir to coat the pecans well and cook for a few minutes until the sugar dissolves. Transfer to a parchment-lined baking sheet to cool completely. Once cooled, break apart any clumped pecans.
Roll out the puff pastry sheet on a lightly floured surface to smooth out any creases. Transfer the pastry to the prepared baking sheet. Score a 1-inch border around the dough with a paring knife, being careful not to cut all the way through. Poke several small holes in the center to prevent it from puffing up during baking. Pre-bake in the oven for 10 minutes or until the tart has risen and is starting to brown. Remove from the oven.
Spread the whipped goat cheese and cream cheese mixture evenly over the puff pastry, leaving the 1-inch border. Layer the par-roasted Sweet Petites carrots in a single layer over the cheese mixture. Brush the pastry border with the beaten egg wash.
Place the tart in the preheated oven and bake for 25-35 minutes, or until the pastry is puffed and golden brown.
Remove the tart from the oven and let it cool for a few minutes before slicing. Garnish with the candied pecans and fresh thyme leaves, if desired.