A machine for processing a liquid food substance, such as milk or a milk-based substance, comprising: - a jug having a container and a handle (31,32,33) for holding the jug by an adult human hand, the container delimiting a food cavity for containing said liquid food substance, such as a container having a removable lid for covering the cavity; and optionally - a base that has one or more walls forming a seat for removably receiving the container, the food cavity containing a movable agitating device, such as a whisk and/or a stirrer, to agitate said liquid food substance during its processing, the handle (31,32,33) having a connection member projecting from the container and a substantially elongated upright gripping member that is connected to the connection member at a distance from the container, the gripping member having an upper portion that is above the connection member and a lower portion that is below the connection member, the upper portion being configured to support a thumb of said hand and delimiting with the connection member an upper seat for accommodating an index finger of said hand when the handle (31,32,33) is held by said hand, optionally: - the connection member being horizontal or at an angle of less than 30 deg thereto, such as less than 15 deg. thereto, when the jug is oriented for processing said liquid food substance; and/or - the upright gripping member extending vertically or at an angle of less than 30 deg thereto, such as less than 15 deg. thereto, characterised in that the handle (31,32,33) has a confinement member that extends: - substantially uprightly between the container and the lower portion of the gripping member; and - downwardly from the connection member or from close thereto, the confinement member and the lower portion and the connection member delimiting a lower seat for accommodating a portion of a middle finger of said hand when the handle (31,32,33) is held by said hand. - a jug having a container and a handle (31,32,33) for holding the jug by an adult human hand, the container delimiting a food cavity for containing said liquid food substance, such as a container having a removable lid for covering the cavity; and optionally - a base that has one or more walls forming a seat for removably receiving the container, the food cavity containing a movable agitating device, such as a whisk and/or a stirrer, to agitate said liquid food substance during its processing, the handle (31,32,33) having a connection member projecting from the container and a substantially elongated upright gripping member that is connected to the connection member at a distance from the container, the gripping member having an upper portion that is above the connection member and a lower portion that is below the connection member, the upper portion being configured to support a thumb of said hand and delimiting with the connection member an upper seat for accommodating an index finger of said hand when the handle (31,32,33) is held by said hand, optionally: - the connection member being horizontal or at an angle of less than 30 deg thereto, such as less than 15 deg. thereto, when the jug is oriented for processing said liquid food substance; and/or - the upright gripping member extending vertically or at an angle of less than 30 deg thereto, such as less than 15 deg. thereto, - the connection member being horizontal or at an angle of less than 30 deg thereto, such as less than 15 deg. thereto, when the jug is oriented for processing said liquid food substance; and/or - the upright gripping member extending vertically or at an angle of less than 30 deg thereto, such as less than 15 deg. thereto, characterised in that the handle (31,32,33) has a confinement member that extends: - substantially uprightly between the container and the lower portion of the gripping member; and - downwardly from the connection member or from close thereto, - substantially uprightly between the container and the lower portion of the gripping member; and - downwardly from the connection member or from close thereto, the confinement member and the lower portion and the connection member delimiting a lower seat for accommodating a portion of a middle finger of said hand when the handle (31,32,33) is held by said hand.