Wash the frogs' legs, then marinate them in the refrigerator for two hours in a food storage bag with the buttermilk and the Tabasco.
For the persille sauce, heat the olive oil in a small saucepan over medium-low heat. Add the garlic and parsley. Cook until the parsley is wilted and the garlic is fragrant. Remove from the heat. Scoop the pan contents into a small food processor, blender, or (most effective) mortar and pestle. Add 1/4 tsp. salt and puree the mixture. Spoon the mixture into one corner of a small plastic bag (like a sandwich bag).
Combine the flour, salt, white pepper and thyme in a wide bowl. Shake the excess buttermilk off the frogs' legs. Coat them lightly with the flour mixture.
Heat the butter in a medium skillet over medium-high heat. When it's melted, add the crushed garlic clove. When the butter is bubbling, add the frogs' legs and sauté until golden, turning once. Remove and drain the frogs' legs. Whisk in the vinegar.
Place the frogs' legs on serving plates. Spoon the butter from the pan onto the plates, trying to avoid picking up the solids on the bottom of the pan.
With scissors, snip off the corner of the plastic bag with the parsley mixture. Squeeze out lines of the persille across the frog's legs. Garnish with lemon halves.