Preheat oven to 350. To prepare crust: Coat a 9- to 10-inch tart pan with cooking spray.
Place flour, tarragon and 1/2 teaspoon salt in a food processor; pulse to combine.
Add butter one piece at a time, pulsing once or twice after each addition, until incorporated.
Add 1/4 cup oil and 1/4 cup yogurt and pulse just until the dough starts to come together.
Transfer the dough to the prepared pan (it will be crumbly), spread evenly and press firmly into the bottom and all the way up the sides to form a crust.
Bake the crust until set but not browned, about 15 minutes. Let cool on a wire rack.
To prepare filling: Heat 2 tablespoons oil in a large skillet over medium heat.
Add onion and cook, stirring occasionally, until tender, about 5 minutes.
Stir in carrots and 1 tablespoon wine or vinegar and cook, stirring, for 2 minutes. Remove from the heat.
Spread mustard over the crust. Sprinkle with Cheddar cheese and then evenly spread the carrot mixture in the tart shell.
Whisk 1/2 cup yogurt, milk, eggs, tarragon, the remaining 1 tablespoon wine or vinegar), 1/4 teaspoon salt and pepper in a medium bowl.
Place the tart pan on a baking sheet and pour in the filling.
Bake the tart until the filling is firm and the edges are golden brown, 40 to 45 minutes.
Let cool for 15 minutes before slicing. Serve warm or chilled.
