Heat a dutch oven or pot over medium high heat and add the olive oil and butter. Season the chicken with the salt and pepper then add it to the pot. Cook until the chicken is golden and browned, then remove it with a spoon and place it in a bowl.
Add the 3 tablespoons of butter to the pot with the onion. Sprinkle in a bit more salt and pepper. Stir, then cook the onions until slightly soft, about 4 to 5 minutes. Add the potatoes and stir, cooking for another 5 minutes and stirring often. Sprinkle the flour over the onions and potatoes and stir gently to create somewhat of a roux with the vegetables. Cook for 2 to 3 minutes while stirring occasionally.
Slowly stream the chicken stock into the pot while stirring – it will thicken. Stir in the squash, green beans, corn and chicken. Stir in the cream. Taste the mixture and season with salt and pepper if needed – I usually add another sprinkle of each.
Remove the crust from the fridge. Roll the pie crust into a… “rustic” shape – no shape necessary really – until it is about 1/4 inch thick. Place the dough on a parchment-lined baking sheet. Spoon the chicken filling into the center, trying to keep it evenly distributed and folding up the edges as you go. Brush the crust with a few spoonfuls of heavy cream.
Bake the galette for 40 to 45 minutes, until the filling is bubbly and the crust is golden. Remove it from the oven and let it cool slightly before slicing and serving, this will help it set up. Sprinkle with fresh herbs and slice!