You can make the rice balls two weeks ahead and freeze. Bread and fry straight from the freezer.
Bring broth to a simmer in a saucepan. Sauté leek and onion in 2 Tbsp. melted butter in a Dutch oven over medium heat 10 minutes. Add garlic and rice; sauté 2 minutes. Stir in wine; cook 3 minutes. Add 1 cup hot broth; cook, stirring constantly, until liquid is absorbed. Repeat, adding all of broth, 1 cup at a time. Stir in 2 Tbsp. butter, 1 1/2 tsp. salt, and 1/4 tsp. pepper. Spread on a baking sheet; cool 10 minutes.
Sauté mushrooms in 2 Tbsp. melted butter and 1 Tbsp. olive oil in a large skillet over medium-high heat 7 minutes. Add salt and pepper to taste. Remove from heat; stir in rice mixture.
Cut cheese into 1/2-inch cubes. Shape mixture into 1 1/2-inch balls, pressing 1 cheese cube into each rice ball; reshape. Cover and chill 30 minutes to 24 hours.
Stir together flour, 1/2 tsp. salt and 1/4 tsp. pepper. Place eggs in a second bowl; place breadcrumbs in a third bowl. Dredge rice balls in flour mixture; dip in eggs. Roll in crumbs. Chill 10 minutes.
Pour oil to a depth of 5 inches into a large Dutch oven; heat over medium-high heat to 350°. Fry fritters, in batches, 3 minutes or until golden; drain on paper towels. Keep warm in a 250° oven.