Oh, Lucy'd happily ply anyone with baked goods 'til the cows come home. She loves feeding people, though she's got something of an attitude problem (mostly when it comes to her mother).
But a hungry officer at her diner? She'd probably bring him coffee (and maybe a slice of pie) before he could even order. If she decides he's a decent guy by her standards, she'd go out of her way to make him feel welcome.