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Carollo's Italian Grocer & Deli in the City Market would be my first suggestion. I know there's also a booth at the actual city market which has fresh pastas every weekend, but the name escapes me.
http://wtv-zone.com/bobalouie/carollo/index.html
For outstanding gelato... I'd actually recommend hitting up Christopher Elbow's new ice cream / sorbet / gelato outpost Glacé. It is phenomenal.
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City Market
20 E 5th St, Kansas City, MO 64106

