Preheat oven to 375ºF. Warm oil in a skillet over medium-high heat. Sauté garlic for 1 minute. Add beef and sauté until no longer pink, about 4 minutes. Stir in onion, celery and carrots; sauté for 5 minutes.
In a pot, cover potato slices with cold salted water and bring to a boil. Cook until just beginning to soften, 2 to 3 minutes; drain.
Sprinkle flour over mixture in skillet; sauté for 1 minute. Add broth and Worcestershire sauce. Cook, stirring, until sauce starts to thicken, about 2 minutes. Transfer to a 9-inch pie pan; cover with potato slices.
Brush pie with melted butter; sprinkle with salt. Bake until filling is hot and bubbly, about 30 minutes. Turn broiler to high and place pie on a rack about 5 inches from heat source. Broil until potatoes are golden brown, 2 to 3 minutes.
I call myself soup lover, but this pie is absolutely delicious!! It's great just as in the recipe. I sometimes add frozen peas and corn or mushrooms, depending on my cravings at the time. I always make a gigantic double or even triple batch because I want enough to share and to have some leftover to enjoy again (my roasting pan isn't just for turkey)! It's that good!!