Barbecued Shrimp "Blt" Sandwich

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1 teaspoon vegetable oil, plus 3 tablespoons

3 ounces (about 5 large) shallots, peeled and sliced

1/2 cup dry white wine

1 tablespoon Champagne vinegar

1/2 cup heavy cream

1/2 cup (4 ounces) unsalted butter, cut into small pieces

1/2 lemon, juiced

Salt

White pepper

1/2 pound (about 20 small) peeled shrimp

1/2 cup mayonnaise, preferably homemade

12 thin slices sourdough bread

2 cups baby lettuce or 8 nice pieces romaine or iceberg lettuce

1 (1/2 pound) tomato, cut into 12 thin slices

6 ounces (12 thick slices) bacon, cooked and drained

Black pepper