Bring a large pot of salted water to a boil. Fill a large bowl with ice water. Add the haricots verts to the boiling water and cook until al dente, about 2 minutes; drain and transfer to the ice water to cool, about 5 minutes. Drain again, shake off any excess water and drain on paper towels. Set aside.
Combine the garlic and cream in a small saucepan. Bring to a simmer over medium-low heat and cook until reduced by two-thirds, about 25 minutes. Transfer to a blender and puree until smooth; let cool.
Meanwhile, bring the sugar and lemon juice to a boil in a medium saucepan over medium-high heat. Add the quince pieces, reduce the heat to low and simmer until they're tender but still hold their shape, about 30 minutes. Let cool slightly, then strain the quinces.
Slice the shallots into 1/4-inch-thick rounds using a mandoline; separate the pieces into individual rings. Soak the shallots in the buttermilk, 30 minutes. Heat 2 inches vegetable oil in a heavy-bottomed skillet over medium-high heat until a deep-fry thermometer registers 350 degrees F. Drain the shallots and dredge in the flour, then add to the hot oil and fry until golden brown, about 4 minutes. Drain on paper towels. Season with salt.
Spoon the garlic cream onto 6 small plates. Top with the haricots verts, fried shallots and a spoonful of candied quince.