Place the reserved goose parts in a medium pot over medium-high heat. Add a touch of oil to lightly coat the bones, and season with salt. Stir until the bones and parts are golden brown and crispy. Add two shallots and the carrot, sautéing until the vegetables are tender, about 8 minutes. Add the thyme and deglaze the pan with enough cold water to cover everything. Bring to a boil, then adjust the heat to a low simmer. The stock should reduce to about a cup after 1 1/2 hours. Strain it and set aside.